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Designing quantum phases in monolayer graphene Nigge, Pascal Alexander 2019

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Designing quantum phases in monolayer graphenebyPascal NiggeB.Sc. Nanostructure Engineering, Julius Maximilian University of Wu¨rzburg,2011M.Sc. Nanostructure Engineering, Julius Maximilian University of Wu¨rzburg,2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)October 2019c© Pascal Nigge, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Designing quantum phases in monolayer graphenesubmitted by Pascal Nigge in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Physics.Examining Committee:Andrea Damascelli, University of British Columbia (Physics and Astronomy)SupervisorDouglas Andrew Bonn, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberGordon Walter Semenoff, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberJohn Madden, University of British Columbia (Electrical and Computer Engineer-ing)University ExaminerAlireza Nojeh, University of British Columbia (Electrical and Computer Engineer-ing)University ExaminerAlexander Gru¨neis, University of Cologne (Physics)External ExaminerAdditional Supervisory Committee Members:Joshua Folk, University of British Columbia (Physics and Astronomy)Supervisory Committee MemberiiAbstractThe physics of quantum materials is at the heart of current condensed matter re-search. The interactions in these materials between electrons themselves, withother excitations, or external fields can lead to a number of macroscopic quantumphases like superconductivity, the quantum Hall effect, or density wave orders. Butthe experimental study of these materials is often hindered by complicated struc-tural and chemical properties as well as by the involvement of toxic elements.Graphene, on the other hand, is a purely two-dimensional material consistingof a simple honeycomb lattice of carbon atoms. Since it was discovered exper-imentally, graphene has become one of the most widely studied materials in arange of research fields and remains one of the most active areas of research to-day. However, even though graphene has proven to be a promising platform tostudy a plethora of phenomena, the material itself does not exhibit the effects ofcorrelated electron physics.In this thesis, we show two examples of how epitaxially grown large-scalegraphene can be exploited as a platform to design quantum phases through interac-tion with a substrate and intercalation of atoms. Graphene under particular strainpatterns exhibits pseudomagnetic fields. This means the Dirac electrons in the ma-terial behave as if they were under the influence of a magnetic field, even thoughno external field is applied. We are able to create large homogeneous pseudomag-netic fields using shallow nanoprisms in the substrate, which allows us to study thestrain-induced quantum Hall effect in a momentum-resolved fashion using angle-resolved photoemission spectroscopy (ARPES).In the second part, we show how the intercalation of gadolinium can be usedto couple flat bands in graphene to ordering phenomena in gadolinium. Flat bandsiiinear the Fermi level are theorised to enhance electronic correlations, and in combi-nation with novel ordering phenomena, play a key role in many quantum materialfamilies. Our ARPES and resonant energy-integrated X-ray scattering (REXS)measurements reveal a complex interplay between different quantum phases in thematerial, including pseudogaps and evidence for a density wave order.ivLay SummaryQuantum materials are at the forefront of current research in physics and mayhold the key for future technologies and applications. Some of the challenges instudying these materials are complicated crystal structures and toxic chemistry.Graphene, on the other hand, is a single atomic layer of carbon atoms arranged ina honeycomb lattice. Its unique mechanical, electronic, and optical properties havemade it a highly sought after and well-studied material since its discovery in 2004.In this thesis, we present two examples how graphene can be used as a platformto study quantum material properties via tailored interactions with adatoms or sub-strates. The results can pave the way towards the on-demand design of quantummaterials on a technologically relevant platform.vPrefaceChapter 3. The results in this chapter are largely based on the publication ”Roomtemperature strain-induced Landau levels in graphene on a wafer-scale platform”by P. Nigge et al. (2019). The work was a collaborative effort between the groupsof A. Damascelli, S. A. Burke, D. A. Bonn and M. Franz at UBC and the group ofU. Starke at the Max Planck Institute for Solid State research in Stuttgart. P. Niggeand A. C. Qu performed the angle-resolved photoemission spectroscopy (ARPES)experiments and analyzed the ARPES data. P. Nigge, A. C. Qu, E. Ma˚rsell, andG. Tom performed the scanning tunnelling microscopy (STM) experiments and an-alyzed the STM data. E´. Lantagne-Hurtubise and M. Franz provided the theoreticalmodelling, with input from C. Gutie´rrez. S. Link and U. Starke grew the sam-ples and performed the atomic force microscopy (AFM) experiment. P. Nigge,A. C. Qu, M. Zonno, M. Michiardi, M. Schneider, S. Zhdanovich, and G. Levyprovided technical support and maintenance for the ARPES setup. A. Damas-celli, M. Franz, S. A. Burke, D. A. Bonn, and C. Gutie´rrez supervised the project.P. Nigge, A. C. Qu, E´. Lantagne-Hurtubise, and C. Gutie´rrez wrote the manuscriptwith input from all authors. A. Damascelli was responsible for the overall projectdirection, planning, and management.Chapter 4. The results in this chapter are largely based on the publication ”Cor-related electron physics in gadolinium intercalated graphene” by P. Nigge et al.(2019). The work was a collaborative effort between the group of A. Damascelli atUBC and the groups of U. Starke and J. Smet at the Max Planck Institute for SolidState research in Stuttgart. P. Nigge and A. C. Qu performed the ARPES measure-ments with 21.2 eV photon energy. P. Nigge analyzed the ARPES measurementswith 21.2 eV photon energy. S. Link performed and analyzed the ARPES measure-viments at the I4 beamline at the MAX III synchrotron in Lund, Sweden. S. Linkand U. Starke grew and characterized the samples. P. Nigge, A. C. Qu, F. Boschini,and R. J. Green performed the resonant energy-integrated X-ray scattering (REXS)measurements at the Canadian Light Source (CLS) in Saskatoon, Canada. P. Niggeand A. C. Qu analyzed the REXS data. R. J. Green performed the REXS theorycalculations. P. Nigge, A. C. Qu, F. Boschini, M. Schneider, S. Zhdanovich, andG. Levy provided technical support and maintenance for the ARPES setup at theUniversity of British Columbia. P. Nigge wrote the manuscript with input from allauthors. A. Damascelli, U. Starke, and J. Smet supervised the project.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Angle-resolved photoemission spectroscopy . . . . . . . . . . . . 172.2 Scanning tunnelling microscopy . . . . . . . . . . . . . . . . . . 242.3 Resonant energy-integrated X-ray scattering . . . . . . . . . . . . 332.4 Low energy electron diffraction . . . . . . . . . . . . . . . . . . . 382.5 Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 413 Strain-induced Landau levels in graphene . . . . . . . . . . . . . . . 484 Correlated electron physics in gadolinium intercalated graphene . . 76viii5 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149B Conference contributions . . . . . . . . . . . . . . . . . . . . . . . . 152ixList of FiguresFigure 1.1 Structure of graphene . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Electronic band structure of graphene . . . . . . . . . . . . . 3Figure 1.3 The quantum Hall effect in graphene . . . . . . . . . . . . . . 5Figure 1.4 Epitaxial graphene on SiC . . . . . . . . . . . . . . . . . . . 7Figure 1.5 Photo of a graphene on SiC sample . . . . . . . . . . . . . . 8Figure 1.6 Graphene as a versatile platform for quantum physics . . . . . 9Figure 1.7 Evidence for superconductivity in lithium decorated graphene 11Figure 1.8 Unconventional superconductivity in twisted bilayer graphene 12Figure 1.9 Model for Mott insulators . . . . . . . . . . . . . . . . . . . 13Figure 2.1 Comparison of experimental techniques . . . . . . . . . . . . 16Figure 2.2 Electron mean free path in metals . . . . . . . . . . . . . . . 18Figure 2.3 Setup of an ARPES experiment . . . . . . . . . . . . . . . . . 19Figure 2.4 Energy diagram photoemission . . . . . . . . . . . . . . . . . 23Figure 2.5 Fermi edge on gold . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.6 ARPES on monolayer graphene . . . . . . . . . . . . . . . . . 25Figure 2.7 Fermi edge on monolayer graphene . . . . . . . . . . . . . . 25Figure 2.8 Scanning tunneling microscopy setup and energy schematic . 26Figure 2.9 Scanning tunneling microscopy modes . . . . . . . . . . . . . 27Figure 2.10 Scanning tunneling microscopy tip conditioning . . . . . . . . 29Figure 2.11 Ag(111) surface state with STM . . . . . . . . . . . . . . . . 31Figure 2.12 STM on epitaxially grown monolayer graphene on SiC . . . . 32Figure 2.13 Resonant X-ray scattering process . . . . . . . . . . . . . . . 36Figure 2.14 Resonant X-ray scattering geometry . . . . . . . . . . . . . . 37xFigure 2.15 Low energy electron diffraction setup . . . . . . . . . . . . . 39Figure 2.16 Simulation of the low energy electron diffraction (LEED) pat-tern for graphene on SiC . . . . . . . . . . . . . . . . . . . . 40Figure 2.17 LEED pattern for monolayer graphene on SiC . . . . . . . . . 41Figure 2.18 Schematic of Raman spectroscopy process . . . . . . . . . . . 43Figure 2.19 Setup of a Raman spectroscopy experiment . . . . . . . . . . 44Figure 2.20 Resonant Raman process in graphene . . . . . . . . . . . . . 45Figure 2.21 Raman spectroscopy of monolayer graphene on SiC . . . . . 46Figure 3.1 Identification of nanoprisms . . . . . . . . . . . . . . . . . . 49Figure 3.2 AFM height distribution . . . . . . . . . . . . . . . . . . . . . 51Figure 3.3 Graphene layer coverage . . . . . . . . . . . . . . . . . . . . 51Figure 3.4 Substrate-induced strain . . . . . . . . . . . . . . . . . . . . 52Figure 3.5 Comparison of Fourier transforms for different graphene de-formations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.6 Momentum-resolved visualization of Landau levels . . . . . . 55Figure 3.7 Fermi velocity and quasiparticle lifetime from ARPES . . . . . 57Figure 3.8 Fit of Landau levels for the exponent . . . . . . . . . . . . . . 58Figure 3.9 Band structure of multilayer graphene . . . . . . . . . . . . . 59Figure 3.10 Model calculation of strain-induced Landau levels . . . . . . 60Figure 3.11 Evolution of Landau levels with increasing uniform pseudo-magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.12 Determination of the mass term . . . . . . . . . . . . . . . . 64Figure 3.13 Sketch of pseudo-Landau levels with Semenoff mass . . . . . 66Figure 3.14 Calculation of pseudo-Landau levels with Semenoff mass . . . 67Figure 3.15 Model fit with constant mass term . . . . . . . . . . . . . . . 67Figure 3.16 Calculation for a uniform mass distribution . . . . . . . . . . 68Figure 3.17 Measurement of the Raman spot size . . . . . . . . . . . . . 69Figure 3.18 Raman maps of the graphene 2D peak . . . . . . . . . . . . . 71Figure 3.19 Raman line spectrum x-direction . . . . . . . . . . . . . . . . 72Figure 3.20 Raman line spectrum y-direction . . . . . . . . . . . . . . . . 73Figure 3.21 Raman line spectra with fit . . . . . . . . . . . . . . . . . . . 74Figure 3.22 Comparison of experimental and model strain . . . . . . . . . 75xiFigure 4.1 Introduction to Gd-intercalated graphene . . . . . . . . . . . 77Figure 4.2 Angle-resolved photoemission spectroscopy (ARPES) on Gd-intercalated graphene . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.3 Anisotropic mass enhancement . . . . . . . . . . . . . . . . . 81Figure 4.4 Ordering phenomena in Gd-intercalated graphene . . . . . . . 84Figure 4.5 Fitting of peaks in the parallel momentum plane . . . . . . . . 84Figure 4.6 Theoretical X-ray absorption spectroscopy scattering factors . 86Figure 4.7 Comparison of theoretical photon energy-dependent scatteringbehaviours . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.8 Polarization dependence of REXS signal . . . . . . . . . . . . 87Figure 4.9 Possible model of magnetic order in Gd-intercalated graphene 89Figure 4.10 Band folding in Gd-intercalated graphene . . . . . . . . . . . 91Figure 4.11 Pseudogap anisotropy at room temperature . . . . . . . . . . 92Figure 4.12 Pseudogap at the M point . . . . . . . . . . . . . . . . . . . . 92Figure 4.13 Symmetrization of photoemission data . . . . . . . . . . . . . 94Figure 4.14 Fitting of energy distribution curves around ARPES kink feature 95Figure 4.15 Fitting results of the energy distribution curves around the ARPESkink feature . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.16 Simulation of coupling to a mode . . . . . . . . . . . . . . . 96Figure 4.17 Simulation of band folding in Gd-intercalated graphene . . . . 98Figure 4.18 Results of simulation of band folding in Gd-intercalated graphene 99Figure 4.19 Analysis of Lorentzian line fits to momentum distribution curves100Figure 4.20 Predicted phase diagram for highly-doped graphene . . . . . . 101Figure 5.1 Combining twisting and strain in graphene flakes . . . . . . . 104xiiGlossary2DEG two dimensional electron gasAFM atomic force microscopyARPES angle-resolved photoemission spectroscopyBZ Brillouin zoneCCD charge-coupled deviceCLS Canadian Light SourceEDC energy distribution curveFS Fermi surfaceFWHM full width at half maximumLL Landau levelLEED low energy electron diffractionLEEM low energy electron microscopyMDC momentum distribution curveMLG monolayer grapheneQED quantum electrodynamicsREXS resonant energy-integrated X-ray scatteringxiiiRIXS resonant inelastic X-ray scatteringSTM scanning tunnelling microscopyUHV ultra high vacuumVHS Van Hove singularityWKB Wentzel-Kramers-BrillouinXAS X-ray absorption spectroscopyXMCD X-ray magnetic circular dichroismXPS X-ray photoelectron spectroscopyXRD X-ray diffractionZLG zero layer graphenexivAcknowledgmentsA PhD thesis in experimental physics is never just the work of a single person.Here I would like to acknowledge all the people who have helped to make thisproject possible.First of all my supervisor Andrea Damascelli, who provided me with the op-portunity to work on an exciting research project, gave me the academic freedom topursue ideas and helped me navigate the sometimes rough waters of modern con-densed matter research. My thesis committee and examiners for taking time out oftheir busy schedules to help evaluate the presented work. The research associatesGiorgio Levy and Sergey Zhdanovich for always trying to keep the laboratory insmooth and operating conditions. A special thanks to our technicians MichaelSchneider and Doug Wong, who always had an open ear and were always willingto lend a helping hand in acquiring, designing, assembling, commissioning, main-taining, and operating new and old equipment in the laboratory. The entire UBCARPES group for support and input during the experimental efforts, especiallyAmy Qu for her time spent with the ARPES, LEED, STM, REXS, and Raman setups.Ulrich Starke and Stefan Link from MPI Stuttgart for growing and character-izing the samples without which none of the presented work would have been pos-sible. Marcel Franz and E´tienne Lantagne-Hurtubise for their theory support andenlightening discussions during the pseudomagnetic field project. The groups ofSarah Burke and Doug Bonn for making physical, human, and temporal resourceson their STM experiments available for our projects. Erik Ma˚rsell and Miriam De-Jong for the preparation of the dogbone samples used for tip conditioning. Thankyou to Brian de Alwis for providing and maintaining the LATEX template.xvAnd last but not least to my parents. Without their continued support and pa-tience this thesis would not have been possible.xviChapter 1IntroductionGraphene is a purely two-dimensional material. It consists of a single atomic layerof carbon atoms arranged in a honeycomb lattice. Due to graphene’s remarkablemechanical, optical, and electronic properties, it is part of a highly active area ofresearch in a number of fields in recent years. As a result, a range of excellentreview articles have been published on the subject of graphene [1–5]. Hence, inthis section only a brief overview summarizing the historical background and themost important properties, as they are relevant to the research of this thesis, aregiven.The existence of free-standing two-dimensional crystals was already discussedover 80 years ago by Landau and Peierls. They argued that such structures couldnot exist because of divergent thermal fluctuations [6, 7]. Therefore, atomicallythin layers were only known as part of heterostructures with lattice matched crys-tals [8, 9]. Nevertheless, the unusual electronic properties of graphene in the con-text of other carbon allotropes were noted early on by theorists [10–12]. Later itwas realized that graphene could also be used as a platform for studying quan-tum electrodynamics (QED) [13–15]. This is due to the fact that charge carriers ingraphene around the Dirac point behave as massless chiral Dirac fermions. This,in principle, allows one to probe relativistic QED phenomena in a condensed mattermaterial at an effective speed of light of vF ≈ 1×106 m/s (300 times smaller thanthe speed of light in vacuum). Experimentally, the advent of graphene began withthe groundbreaking discovery of Geim and Novoselov and the subsequent confir-1Figure 1.1: Structure of graphene. (a) The carbon honeycomb lattice of graphenecan be described as two triangular lattices A (red) and B (blue). The unit cellvectors are denoted by a1−2 and the three vectors connecting nearest neighboursby δ1−3. (b) Illustration of the overlap between the in-plane sp2 hybridized orbitalsof adjacent carbon atoms. These form the so called σ bonds in graphene. (c) Theout-of-plane pz orbitals, in contrast, form the so called pi bonds, that appear similarto the delocalized electron systems of aromatic molecules for example.mation of the Dirac nature of the charge carriers in graphene [16–19]. Since then,an ever-growing number of research papers have been published and graphene stillholds promise for a range of applications from flexible electronics to energy storageand biomedical applications [20–22].The structure of graphene is illustrated in Figure 1.1a. The crystal can be de-scribed as two triangular lattices which together form the honeycomb structure.The carbon-carbon distance is a≈ 1.42A˚ and the lattice vectors area1 =a2(3,√3) and a2 =a2(3,−√3). (1.1)The three nearest-neighbour vectors in real space are given byδ1 =a2(1,√3), δ2 =a2(1,−√3) and δ3 =−a(1,0). (1.2)Carbon atoms have the electronic configuration [He]2s22p2. In graphene, two ofthe p orbitals and the s orbital sp2 hybridize and form three in-plane orbitals withan angle of 120◦ between them. Each of them overlaps with an in-plane orbital ofa neighboring carbon atom. These are the so called σ bonds (see Figure 1.1b). Theremaining out-of-plane pz orbitals contain one electron per carbon atom and form2Figure 1.2: Electronic band structure of graphene. (a) Schematic of the Bril-louin zone (BZ) of graphene with high symmetry points and reciprocal lattice vec-tors b1−2 labelled. (b) Band structure of graphene as calculated using a tight-binding approach. For free standing graphene the bands are half filled, leading toa Fermi surface (FS) with six points at the corners of the Brillouin zone (BZ). (c)Linear dispersion around the K point forming a Dirac cone. (d) Energy contoursof the graphene band dispersion. Starting from the Dirac point, electronic dopingleads to a transition from a circular FS to a trigonally warped FS.the so called pi bonds (see Figure 1.1c). Similar to many aromatic molecules, theelectrons in the pi bonds are highly delocalized (compare for example benzene).Many of the remarkable properties of graphene originate from the electronicband structure close to the Fermi level. The bands close to the Fermi level are dic-tated by the out of plane pi bonds. The bonding and antibonding bands originatingfrom the σ bonds lay further away from the Fermi energy. Graphene’s Brillouin3zone (BZ) in reciprocal space is depicted in Figure 1.2a. Important high symmetrypoints in the hexagonal BZ around the Γ point are K and K’ at the corners and Mat the center along the edges. Their locations in momentum space as depicted inFigure 1.2a are as follows:K = (2pi3√3a,2pi3a), K′ = (4pi3√3a,0) and M = (√3pi3a,pi3a). (1.3)The reciprocal lattice vectors are given byb1 = (2√3pi3a,2pi3a) and b2 = (0,4pi3a). (1.4)The electronic band structure can be approximately calculated using a tight bindingapproach [10, 23]. Using the lattice constant a0 = 2.46A˚, the nearest neighborhopping t (≈ 2.8eV) (between different sublattices) and the next nearest neighborhopping t ′ (between the same sublattices), the resulting dispersion isE± =±t√3+ f (k)− t ′ f (k) withf (k) = 2 cos(kx a0)+4 cos(kxa02) cos(√3kya02).(1.5)Here the “+” sign stands for the upper (pi∗) band and the “−” sign for the lower piband. Note, if t ′= 0 is assumed as an approximation, the dispersion is particle-holesymmetric around zero energy. The resulting bands for that case are displayed inFigure 1.2b. The upper and lower bands meet in singular points at the corners of theBZ, forming the iconic linear dispersing Dirac cones (see Figure 1.2c). The Fermivelocity can be obtained by expanding the band structure around the K points,yielding vF = 3ta2h¯ ≈ 1×106 m/s [10]. At the M points the dispersions of two neigh-bouring Dirac cones merge, forming a highly anisotropic saddle point in the bandstructure. Looking at iso-energy contours as the energy is changed starting fromthe Fermi level, the circular electron pockets around the K and K’ points begin toshow trigonal warping along the high symmetry directions. Eventually the pocketsmerge at the M points and form a single hole pocket centered around the Γ point.After the discovery of single-layer graphene, the quantum Hall effect played acrucial role in the confirmation of the chiral Dirac nature of the charge carriers in4Figure 1.3: The quantum Hall effect in graphene. (a) Quantum Hall effect asit would present itself in the longitudinal conductivity σ as a function of chemicalpotential for a two dimensional electron gas (2DEG). The Landau levels (LLs)have a linear succession with no peak at zero energy. (b) Quantum Hall effect forDirac electrons in graphene. The LLs follow a square root sequence with a LL atzero energy. (c) For bilayer graphene the linear series recovers, but the LL at zeroenergy is still present. All LLs are broadened to reflect disorder and are invertedfor hole carriers for clarity.the material [17, 19]. The quantum Hall effect is a remarkable macroscopic phe-nomenon [24–26] which was first observed by K. v. Klitzing in 1980 [27]. It laidthe ground work for the now very large community of topological order in con-densed matter [28–30]. The effect is typically observed in clean two-dimensionalmaterials under large magnetic fields and at cryogenic temperatures. Under theinfluence of the magnetic field, the charge carriers are forced onto quantized cy-clotron orbits called Landau levels (LL). This leads to the famous plateaus in theHall conductivity as the chemical potential is changed. This quantization ( he2 ) onlydepends on natural constants and can thus be used as a precise standard referencefor the electrical resistance [31, 32]. In normal two-dimensional electron gases(2DEGs) the quantization is linear in the LL index N and no level lies at zero en-ergy (see Figure 1.3a). In contrast to that, LLs in graphene follow a√N behaviorwith a prominent signal at zero energy (see Figure 1.3b). This can serve as a hall-mark of Dirac electrons in graphene for which the energetic position of the LLs canbe described by [33–36]EN =±vF√2eh¯BN. (1.6)5Here “±” refers to hole- or electron-like charge carriers in the upper or lower coneof the dispersion, vF is the Fermi velocity, and B is the magnitude of the perpendic-ular magnetic field. Because of the comparatively large energy spacings betweenthe LLs even in moderate magnetic fields and the long lifetimes of the charge car-riers between scattering events, the quantum Hall effect in graphene can even beobserved in room temperature environments [37]. It thus marked the first macro-scopic quantum phenomenon – outside the well-known areas of magnetism andsemiconductor physics – to be present at nearly ambient conditions. For complete-ness the succession of LLs in bilayer graphene is depicted in Figure 1.3c. The usuallinear dependence is recovered, but the state at zero energy still exists.For any experimental effort, it is crucial to have access to clean and high qualitysamples. In the case of graphene, there are currently a number of approaches beingcarried out, all with their own advantages and disadvantages. They can in generalbe divided into two categories: top-down approaches in which three-dimensionalgraphite is exfoliated down to the two-dimensional graphene limit, or bottom-upapproaches in which graphene is grown on substrates. The original discovery bythe Manchester group relied on the top-down technique of mechanical exfoliation(also known as the Scotch tape method) [17, 38–40]. It produces high-qualitysamples, but only yields small quantities and is not very reproducible. Other top-down methods are based on the exfoliation of graphite in solution, which allowsthe production of graphene on industrial scales, but yields less high-quality sam-ples [41–45]. On the bottom-up side, graphene can be directly synthesized fromorganic precursor molecules [46–50] or catalyzed to grow on substrates [51–56].Producing large and uniform monolayers of graphene remains challenging for bothapproaches.Epitaxially grown graphene on SiC is considered a promising route for the pro-duction of high-quality graphene on an insulating substrate on a wafer-scale size[57–59]. SiC is a large band gap (≈ 3eV) semiconductor and exists in a number ofdifferent crystal structures. The graphene samples used for the research presentedin this thesis were grown on 6H-SiC, which has a hexagonal lattice structure with alayered stacking order of ABCACB (see Figure 1.4a). The commercial wafers (seeFig. 1.5) are cleaned by hydrogen etching prior to the growth process. Grapheneis then grown via the sublimation of silicon on the silicon-terminated surface of6Figure 1.4: Epitaxial graphene on SiC. (a) Schematic of the structure of the 6H-SiC substrate with its ABCACB stacking order. The top view (bottom) shows thethree different layer-dependent registries for the Si atoms. (b) The first graphiticlayer grown on the Si-terminated substrate saturates the Si dangling bonds and isthus not free-standing graphene. It is often called zero-layer graphene (ZLG) (top).A second carbon layer can be grown keeping the ZLG in place as a buffer. The toplayer then acts as monolayer graphene (MLG) (middle). Alternatively, the Si bondscan be saturated through the intercalation of hydrogen also leading to a quasi freestanding layer of graphene. (bottom). (c) Due to the lattice mismatch between theSiC substrate (yellow) and the graphene (blue) a reconstruction emerges, in which13×13 graphene cells fit (6√3×6√3)R30◦ cells of the SiC to 0.1%.the substrate at about 1500◦C under an argon atmosphere of about 900 mbar. Thefirst layer is the so called buffer layer (sometimes also called zero-layer graphene –ZLG). The buffer layer structurally already resembles graphene, but does not showa Dirac cone at the corners of the BZ. This is due to the dangling Si bonds at thesurface. They bind to 13 of the carbon atoms in the buffer layer, thereby destroy-ing the pi bands of graphene [60]. To establish a quasi free standing monolayer ofgraphene, two approaches are possible: first, a second carbon layer can be grownleading to an electron-doped monolayer graphene sample which is decoupled fromthe substrate through the buffer layer. Alternatively, the dangling Si bonds can besaturated via the intercalation of hydrogen between the graphene and the substrate,which leads to a hole-doped monolayer graphene sample (see Figure 1.4b) [61].Because of the lattice mismatch between graphene and the SiC substrate, a recon-struction emerges. 13×13 graphene cells fit (6√3×6√3)R30◦ cells of the SiC to0.1%, which is illustrated in Figure 1.4c [59, 60]. One drawback of graphene on7Figure 1.5: Photo of a graphene on SiC sample. A typical graphene on SiCsample with a size of 10 mm × 20 mm is shown (red). The wafers (commerciallybought from SiCrystal GmbH) are about 350 µm thick and are doped with nitro-gen atoms, but nevertheless become insulating below about 50 K. The samples aremounted using a copper paste (blue), which guarantees good thermal and electricalcontact of the graphene. The specifics of the sample holder (yellow) depend onthe experiment. Here shown is the sample holder for the X-ray scattering experi-ments. On both sides the shielding of the cryostat is visible (green), which preventsthermal radiation from heating up the sample.SiC is the existence of terraces steps on the substrate due to slight misalignmentof the wafer cuts from the ideal crystal orientation. This leads to small contribu-tions of bi- and trilayer graphene during the growth process [57]. Epitaxially grownmonolayers of graphene have already been used extensively in research, as they arecompatible with a range of experimental techniques and offer the flexibility of, forexample, adjusting the doping level and studying many-body physics phenomenathrough the intercalation or addition of suitable atoms [62–66].Combining graphene’s properties with the availability of a plethora of exper-imental data in a abroad range of research fields as well as high quality samples,graphene is positioned to be a unique candidate as a platform for the on-demanddesign of quantum phases. The theoretical idea of using graphene to simulatephysics from different research fields was pointed out early by G. W. Semenoff in1984 [13], but experimentally it took until 2004, when research groups around8Figure 1.6: Graphene as a versatile platform for quantum physics. Schematicillustrations of different tuning parameters and control knobs which can be appliedto graphene to induced quantum phases on demand. (a) Atoms can be added ontop of graphene or intercalated in between the graphene and the substrate. (b)Multiple graphene layers can be coupled and twisted with respect to each other. (c)Graphene layers can be strained or compressed. (d) Graphene nanoribbons can besynthesized, further reducing the dimensionality of the material. (e) Spin relatedproperties and magnetism can be induced by adding magnetic moments. (f) Ultra-fast light pulses allow the manipulation and creation of novel quantum phases.K. S. Novoselov and A. K. Geim were able to isolate single layers of graphene andconfirm the Dirac nature of their charge carriers [16–18]. Soon it was noted thatfor the exploitation of graphene’s properties in optical or electrical applications liketransistors, it would be necessary to open a band gap in the electronic structure. Thequest to engineer such a gap became and still is the first showcase of how electronicproperties could be designed starting with graphene as the platform. Until todaya range of approaches to open a band gap have been proposed and experimentallydemonstrated and the search still continues [67–72].9An overview of the tuning parameters and control knobs available for grapheneis illustrated in Fig. 1.6. The first method is adding atoms to graphene (see Fig. 1.6a).This can happen as a decoration on top of graphene or as an intercalation betweenthe graphene sheet and the supporting substrate. Depending on atomic species,deposition amounts and flux, substrate, as well as temperature, the arrangementand adsorption sites with respect to graphene can be varied. One particular ex-ample was the addition of alkali atoms to graphene. It was theoretically pre-dicted that graphene decorated with a superlattice of lithium atoms can enhancethe coupling between electrons and phonons in the system and hence lead to theemergence of superconductivity [73]. Experimentally, such an enhancement ofthe electron-phonon coupling was indeed observed in angle-resolved photoemis-sion spectroscopy (ARPES) measurements after in situ deposition of different al-kali atoms [62]. In 2015, it was again an ARPES effort that found evidence for asuperconducting gap in the band structure of graphene, following the depositionof lithium atoms at cryogenic temperatures [66]. The main findings are summa-rized in Fig. 1.7. Upon deposition of the lithium atoms, an electronic doping isobserved due to a transfer of electrons from lithium to graphene, shifting the Diracpoint down to a binding energy of about 700 meV. Further, as long as the sampleis kept at liquid helium temperatures during and after the deposition, the lithiumatoms indeed appear to be ordering in the theoretically suggested superstructure.This is apparent in the Fermi surface (see Fig. 1.7b) by a back folding of the Diraccones across the new Brillouin zone boundaries to the center of the zone. Addition-ally, the dispersion of the graphene bands shows prominent kink features near theFermi level (see Fig. 1.7c). These deviations from the expected linear dispersioncan be used to extract the electron-phonon coupling parameter λ . It was shownthat an enhancement of the coupling between electrons and phonons could indeedbe observed and that the primary contribution stems from lower energy vibrationsin the material (see Fig. 1.7d). Finally, a temperature-dependent gap was observedafter the deposition of lithium, suggesting a superconducting phase with a criticaltemperature of Tc ' 5.9K (see Fig. 1.7e).Another tuning parameter that has recently led to a frenzy of experimentaland theoretical efforts is the twisting between two or more graphene layers (seeFig. 1.6b). At certain twist angles (so-called “magic angles”) the induced Moire´10Figure 1.7: Evidence for superconductivity in lithium decorated graphene. (a)ARPES cut through the Dirac cone as indicated by the white line in (b). The depo-sition of lithium induces an electronic doping, shifting the Dirac point to a bindingenergy of about 700 meV. (b) Fermi surface of lithium-decorated graphene. Dueto the superlattice of the lithium atoms on graphene, the size of the Brillouin zonechanges (solid white – original and dashed white – new), leading to a folding ofbands to the Γ point at the center of the Brillouin zone. (c) A close up ARPEScut of one of the Dirac cone branches reveals kink features – indicative of electron-phonon coupling – near the Fermi level. (d) The electron-phonon coupling parame-ter λ extracted as a function of Li deposition time. The contribution of high-energyphonon modes (white circles) remains almost constant, while the electron-phononcoupling for lower-energy vibrational modes (black circles) increases significantly.(e) ARPES energy distribution curves (EDCs) at the point indicated by the whitecircle in (b) show a temperature-dependent gap opening between 15 K (red) and3.5 K (blue). The data was symmetrized with respect to the Fermi energy. Thisfigure is adapted from [66].11Figure 1.8: Unconventional superconductivity in twisted bilayer graphene. (a)Simulated band structure for twisted bilayer graphene at an angle of 1.05◦. Flatbands emerge around the center of the Brillouin zone. (b) Current-voltage (Vxx− I)curves for a twisted bilayer graphene device at various temperatures with a car-rier density of n =−1.44×1012 cm−2. The lowest temperatures show a plateau ofvanishing resistivity with a critical current of about 50 nA. (c) Resistance versustemperature curves (Rxx−T ) for different magnetic fields B⊥. The region of zeroresistance vanishes with increasing magnetic field. (d) Phase diagram of the resis-tance (Rxx) in a twisted bilayer graphene device as a function of carrier density andtemperature. Two superconducting domes are flanking a proposed Mott insulatingphase, resembling the phase diagrams for other unconventional superconductorslike the cuprates. This figure is adapted from [74].pattern leads to the appearance of flat bands at the Fermi level (see Fig. 1.8a). Flatbands in general are associated with correlated electron behaviour as the screeningof the charge carriers is greatly reduced and interactions between individual elec-trons become important. In a hand waiving picture, one can think of a situationwhere the charge carriers’ effective mass is going to infinity and hence their abilityto screen fluctuations is reduced.Historically, the effects of electron correlations in solids were first observed intransition metal oxides. Band theory predicts a fully filled oxygen p-band (lightgrey in Fig, 1.9) and a partially filled transition metal d-band (dark grey and whitein Fig, 1.9) in these materials, hence giving them metallic character. In contrast tothis expectation many compounds showed insulating behaviour instead, indicatinga band gap at the Fermi level [75]. Nevill Mott and Rudolf Peierls proposed apicture, based on the interactions between electrons, explaining these observations.On the one hand, when electrons are treated independently, the kinetic energy of12Figure 1.9: Model for Mott insulators. Two contrasting pictures for the densityof states (DOS) around the Fermi level (EF ) are depicted. (a) In the independentelectron picture a band with width W is crossing the Fermi level. The partially filledband leads to metallic behaviour. (b) If correlations between electrons becomeimportant, the Coulomb repulsion energy U can open a gap at the Fermi level. Theresult is insulating behaviour even though band theory would predict a metal.the charge carriers dominates and the hopping of electrons from atom to atomleads to an energy band of width W . Metallic behaviour is observed (compareFig. 1.9a). On the other hand, the Coulomb repulsion energy U is included. Uindicates the cost of an electron hopping into an atomic orbital that already hasan electron in it. If the Coulomb repulsion is sufficiently large compared to thekinetic energy, hopping to the next site is suppressed and the material becomes aninsulator (compare Fig. 1.9b) [76, 77]. This picture later led to the development ofthe Hubbard model which is still of high interest today as it is thought to describethe essential physics of high-temperature superconductors [78, 79].The importance of electronic correlations in graphene has been discussed ex-tensively in theory [1, 80, 81] and indeed effects of many body interactions havebeen found experimentally, especially as the material is doped away from the Diracpoint [64, 66, 82]. Experimental efforts in search for correlated electron behaviourin graphene culminated in recent findings showing the emergence of an unconven-tional superconducting phase in magic angle twisted bilayer graphene devices attemperatures below 1.7 K [74]. Transport measurements show a zero resistancestate (see Fig. 1.8b) that is suppressed with the application of magnetic fields (seeFig. 1.8c). Mapping out the phase diagram of the resistance with the carrier den-13sity and the temperature as parameters (see Fig. 1.8d), reveals two superconductingdomes adjacent to an unexpected insulating state [83]. The phase diagram showsintriguing similarities to the ones typically found in the high-temperature super-conductors such as the cuprates [84].Another powerful control knob for the platform of graphene is strain or pres-sure to engineer the electronic band structure of graphene (see Fig. 1.6c). Oneexample is the generation of pseudomagnetic fields under certain strain geometries[85]. The Dirac electrons in graphene behave as if they were in a magnetic fieldand can quantize into flat Landau levels, even though no external magnetic field hasbeen applied. Pseudomagnetic fields were first observed in so called nanobubblesusing scanning probe techniques and, depending on the magnitude of the strain,can reach several hundred tesla in strength [86]. The next parameter is a furtherreduction of dimensionality in graphene, leading to the formation of nanoribbons(see Fig. 1.6d). These can host interesting edge states depending on the edge ge-ometry of the flake [67, 68, 87, 88]. Further, the prospect of adding spin-relatedproperties to graphene has been studied extensively (see Fig. 1.6e). Magnetism canbe introduced to graphene by adding atoms [89], proximity effects [90], or evendefects [91]. Additionally, despite carbon being one of the lighter atoms in theperiodic table with a small intrinsic spin-orbit coupling, it can be induced by theaddition of Pb atoms to the carbon lattice of graphene [92, 93]. Finally, the ex-ploitation of novel ultrafast pump-probe techniques can be a powerful tuning knob(see Fig. 1.6f). Dirac carriers can be studied out of equilibrium in the time domain[94] and experiments inducing ultrafast photocurrents in topologically insulatingmaterials [95] should soon become possible in graphene as well.This shows that graphene can be a versatile platform with a number of availabletuning parameters to study the design of quantum phases. After introducing theimportant basic concepts of the experimental techniques in the next chapter, twoexamples of how specific control knobs in graphene can be used to study novelquantum phases will be presented in this thesis.14Chapter 2MethodsIn this chapter, the experimental techniques used during the course of the researchwill be briefly described. They can roughly be divided into three categories de-pending on whether they primarily provide information in real space, reciprocalspace, or the BZ in momentum space (see Fig. 2.1). All of them have their ownstrengths and weaknesses in general and regarding the specific experiments withgraphene on SiC in this thesis.The experimental technique of STM is based on the precise scanning of anatomically sharp tip across the sample. If the distance between the tip and thesample is small enough (∼ fewA˚), charge carriers can tunnel from the tip into thematerial or vice versa depending on the applied voltage. STM thus can provideimages with atomic resolution, while at the same time giving spectroscopic accessto the local density of states by sweeping the bias voltage. Raman spectroscopy, onthe other hand, is based on the inelastic scattering of monochromatic light. Photonsfrom a laser source are focused onto the sample, which can be scanned laterally.The photons can couple to excitations in the material, giving rise to a characteristicshift in the energy of the outgoing photon. The reflected light is then analyzed in aspectrometer.To study ordering phenomena in a material, scattering techniques are ideallysuited. Also here different particles can be used as scattering probes. For LEEDmonochromatic electrons (in this thesis typically between 10 eV and 100 eV) arefocused onto the sample. The diffracted electrons are then detected in reflection15Figure 2.1: Comparison of experimental techniques. (a) Scanning tunnelingmicroscopy (STM) can provide spectroscopic information in real space with upto atomic resolution. Raman spectroscopy allows the measurement of excitationsin a material, but the lateral resolution is limited to the size of the laser spot onthe sample (∼ 1µm) (b) Scattering techniques like low-energy electron diffraction(LEED) or resonant energy-integrated X-ray scattering (REXS) provide direct ac-cess to ordering phenomena in reciprocal space. (c) Angle-resolved photoemissionspectroscopy relies on the emission of electrons from the material. It is conceptu-alized best using the Brillouin zone (BZ) in momentum space.on a phosphor screen. In REXS a similar principle is applied, but monochromaticX-ray photons instead of electrons are used. This experiment is carried out atsynchrotron facilities which allows tuning of the photon energy (typically between100 eV and 2000 eV for soft X-ray storage rings). While doing so, resonancesfor specific absorption edges can be selected. This enables chemical- and orbital-selective measurements with a high signal-to-noise ratio, allowing the detection ofshort-range ordering phenomena. As a plus, the additional degree of freedom ofphoton polarization can be used to distinguish between spin and charge excitations.Finally, ARPES relies on the photoelectric effect. A monochromatic incom-ing photon emits a photoelectron, which carries information about its binding en-ergy (kinetic energy of photoelectron) and crystal momentum (emission angle ofphotoelectron) inside the material. Analyzing both, ARPES allows the direct mea-surement of the spectral function in the BZ for charge carriers below the Fermilevel. Over the years, many improvements and extensions to the “standard” ARPESexperiments have become available, pushing for better energy and angular reso-lution, allowing for the detection of the photoelectron’s spin, and implementing16laser-based pump-probe setups with ultra short light pulses for measurements inthe time domain.2.1 Angle-resolved photoemission spectroscopyARPES is a powerful technique for the study of the electronic properties of quantummaterials. It has broad applications in a range of scientific fields and has signifi-cantly contributed to our current understanding of the underlying principles in solidstate physics. In this section, a brief overview of the historic and scientific back-ground of the experimental method is given. For a more in-depth overview, sev-eral excellent books [96–101] and review articles [102, 103] are available. ARPESis based on the photoelectric effect, which was first explored in experiments byH. Hertz and W. Hallwachs in 1887 [104, 105]. The theoretical groundwork waslaid in one of A. Einstein’s famous publications in 1905 [106]. It paved the way tothe underlying principle of photoemission as we know it today, in which the kineticenergy of the emitted electron Ekin is related to its binding energy in the materialEB, the energy of the incoming photon hν , and the work function of the materialφ .Ekin = hν−φ −|EB| (2.1)Experimentally, this means that samples are exposed to monochromatic lightand the emitted electrons are analyzed according to their kinetic energy. Photonsources routinely used for photoemission experiments are helium gas dischargelamps (He I: 21.2 eV and He II: 40.8 eV) and lasers in laboratory-based settings,or large-scale synchrotron facilities providing tunable photon energies in the softX-ray regime. The resulting kinetic energies for the emitted electrons are often in arange with very short mean free paths (see Fig. 2.2). This means ARPES is a highlysurface sensitive technique, ideally suited to monolayer samples like graphene.However, this also means that samples have to be clean on an atomic level and keptclean during the experiment. Hence, measurements and sample preparation haveto be conducted under ultra high vacuum (UHV) conditions (pressure better than10−10 Torr).For many quantum phenomena, not only is the energy of electrons near theFermi level crucial, but also their dependence on momentum. Therefore, in addi-17Figure 2.2: Electron mean free path in metals. Electron mean free paths as afunction of electron kinetic energy in various materials. Many follow the so called“universal curve” (red line), showing a minimum around 50 eV kinetic energy witha mean free path of only a few A˚, corresponding to only a couple of atomic layersin most samples. Adapted from [107].tion to the kinetic energy of the photoelectron, also the two emission angles (polarangle θ and azimuthal angle ϕ) have to be measured (see Fig. 2.3a). The most com-mon electron analyzers for ARPES are hemispherical analyzers. Electrons emittedfrom the sample are focused onto the entrance slit and accelerated or deceleratedto a preset pass energy in the lens column (see Fig. 2.3b). In the actual hemisphere,the electrons disperse according to their kinetic energy and one of the emission an-gles. Modern analyzers have two dimensional detectors usually consisting of a mi-cro channel plate, a phosphor screen, and a charge-coupled device (CCD) camera.They can measure a range of kinetic energies and emission angles simultaneously.18Figure 2.3: Setup of an ARPES experiment. (a) Geometry of an ARPES experi-ment. An incoming photon (red) hits the sample and emits an electron (blue) underthe polar angle θ and azimuthal angle ϕ . (b) Schematic of a hemispherical elec-tron analyzer. Photoelectrons emitted from the sample enter the lens column andare focused onto the entrance slit of the hemisphere. The electrons disperse accord-ing to their angle theta and kinetic energy Ekin, before hitting the two-dimensionaldetector. (a) adapted from [102].ARPES takes advantage of the fact that the in-plane translational symmetry dur-ing the photoemission process is preserved. Hence, the parallel components of themomentum inside the material k‖ can be directly related to the measured emissionangles and therefore parallel components of the momentum of the photoelectronoutside the material K‖:kx = Kx = 1h¯√2mEkin sinθ cosϕ (2.2)ky = Ky = 1h¯√2mEkin sinθ sinϕ. (2.3)Note here that the momentum carried by the photon can be neglected for typicalenergies used in photoemission experiments. For the perpendicular component ofthe momentum k⊥ the situation is more complicated, as it is not conserved dur-ing the photoemission process. One can nevertheless determine the out-of-planecomponent, if the assumption of a nearly free electron as the final state is made.19The assumption is routinely made in photoemission experiments; however, it is ex-pected to work well only in systems with simple free-electron-like Fermi surfacesand high-energy final states for which the influence of the crystal potential is small.Under this assumption k⊥ can be written ask⊥ = 1h¯√2m(Ekin cos 2θ +V0). (2.4)Here V0 is the so-called inner potential. It can be determined in a number of dif-ferent ways: (i) setting V0 equal to the muffin tin potential from band structurecalculations, (ii) using a V0 that optimizes the agreement between the measuredand expected band structure, and (iii) experimentally by varying the photon energyand observing the periodicity in k⊥ [102]. Note that for two-dimensional mate-rials like graphene, the momentum component perpendicular to the sample is notproperly defined, avoiding this complication altogether.For a formal description of the photoemission process, it is instructive to startwith Fermi’s golden rule, which estimates the transition probability w f i between aninitial state Ψi with energy Ei and a final state Ψ f with energy E f for an incomingphoton with energy hν :w f i = 2pih¯ 〈Ψ f |Hint |Ψi〉2 δ (E f −Ei−hν). (2.5)For a definition of the perturbation operator Hint , we can look at the interactionof an electron with mass m and the electromagnetic field A. Starting from theunperturbed Hamiltonian H0 = p2/2m+ eV (r), this leads to the transformationp→ p− ec A for the momentum operator:H = 12m[p− ec A]2+ eV (r)= p22m +e2mc(A ·p+p ·A)+ e22mc2 A2+ eV (r)= H0+Hint .(2.6)The quadratic term in A is usually neglected, as it only becomes relevant for ex-tremely high photon intensities. Furthermore, if one assumes the electromagneticfield is constant over atomic distances, one can simplify the commutation relation20for A and p: [p,A]=−ih¯∇ ·A = 0. (2.7)Note that the term ∇ ·A can become important when effects like surface photoe-mission are taken into account [108–112]. The resulting Hint is thenHint = emc A ·p. (2.8)An additional simplification in the description of the photoemission process oftenused is the so-called sudden approximation. It states that the emitted photoelectronleaves the sample instantaneously and does not interact with the remaining (N−1)-electron system left behind. This guarantees that the final state can be written asthe product of the emitted electron and the (N−1) wave function. For low kineticenergy electrons, the time to leave the sample might become comparable to theresponse time of the system and the approximation breaks down. One can distin-guish the two extreme cases. In the adiabatic regime the (N− 1) system remainsin its ground state and in photoemission a single symmetric peak is measured. Onthe other hand in the sudden regime, the (N−1) system is in an excited state andin photoemission additional peaks or tails are measured due to the overlap withseveral many-body states. [113–115]When discussing photoemission experiments on correlated electron systems,the Green’s function formalism can be a useful approach [116–121]. Here, themany body interactions can be expressed in terms of the complex self-energyΣ(k,ω) = Σ′(k,ω)+ iΣ′′(k,ω) of an electron with energy ω and momentum k.The real and imaginary parts of the self-energy renormalize the energy and life-time of the bare dispersion εk of the electron. The Green’s function as well asspectral function can both be expressed in terms of this self-energy:G(k,ω) =1ω− εk−Σ(k,ω) (2.9)andA(k,ω) =− 1piΣ′′(k,ω)[ω− εk−Σ′(k,ω)]2+[Σ′′(k,ω)]2 . (2.10)21This implies that the spectral function as measured with ARPES can be directly re-lated to the imaginary part of the Green’s function (A(k,ω) =−( 1pi )Im(G(k,ω))).However, in reality the measured intensity depends on a number of factors and canbe written as follows:I(k,ω) = I0(k,ν ,A) f (ω)A(k,ω). (2.11)Here I0(k,ν ,A) is proportional to the square of the one-electron matrix element andtherefore depends on the momentum of the electron and the energy and polarizationof the incoming photons [102]. Because ARPES removes electrons from the system,only the occupied density of states can be probed and we have to include the Fermifunction f (ω) = (exp(ω/kBT ) + 1)−1. In addition, the effects of a backgroundsignal and resolution broadening in momentum as well as in energy also have to betaken into account.A schematic energy diagram of the photoemission process is illustrated inFig. 2.4. The sample is electrically connected to the analyzer as well as to ground,so that emitted electrons can be replenished to avoid charge build-up and the Fermilevels in the sample and the analyzer are aligned. An incoming photon with theenergy hν emits an electron with kinetic energy E ′kin depending on the electron’sbinding energy Ebin and the work function of the sample φsam.. The kinetic energyEkin that is actually measured at the analyzer is different, as the work function of theanalyzer φana. differs from that of the sample. As the work function of the analyzeris typically not known a priori or may change with time, reference measurementsto determine the exact location of the Fermi level are needed. An example is shownin Fig. 2.5, where a polycrystalline gold sample was used, which can be preparedin situ in the UHV chamber. It shows the expected Fermi cut off which smears outwith increasing temperature. The data can be fitted to a Fermi function after includ-ing an additional convolution with a Gaussian for the finite energy resolution. Thisway, the energetic position of the Fermi level can be established (around 16.85 eVkinetic energy in Fig. 2.5).Now turning to a typical ARPES measurement on monolayer graphene on SiC,a Fermi surface and two cuts are depicted in Fig. 2.6. The samples were glued tothe sample holder using a copper paste, guaranteeing good electrical and thermal22Figure 2.4: Schematic energy diagram for photoemission. The sample is elec-trically connected to the analyzer, so that the Fermi levels in the sample and theanalyzer are aligned. An incoming photon with energy hν emits an electron withkinetic energy E ′kin depending on the electron’s binding energy Ebin and the workfunction of the sample φsam.. The kinetic energy Ekin that is actually measured atthe analyzer is different, as the work function of the analyzer φana. differs from thatof the sample. Adapted from [122].contact, and cleaned in situ in an oven at about 500◦C over night. Due to chargetransfer from the substrate to the graphene, monolayer samples on SiC show a sig-nificant electron doping. This leads to a shift of the Dirac point to about 450 meVbinding energy. The Fermi surface already shows signs of trigonal warping, ex-pected for these doping levels. One can look at the Fermi cut off by integrating inmomentum over one of the linear branches of the Dirac cone, perpendicular to theFermi surface (see Fig. 2.7). This way the resolution for different pass energy set-tings on the sample can be compared as long as the sample temperature is known.Note that these measurements can also be done on the gold calibration sample orany other metallic sample without strong spectroscopic features near the Fermilevel. As expected, the energy resolution worsens with increasing pass energy, butthe transmission of the analyzer is higher with increasing pass energy. Hence, a23Figure 2.5: Fermi edge on gold. Measured Fermi edge on a polycrystalline goldsample at two different temperatures. Data taken at 3.9 K is shown in blue, and10.3 K in red. The data is fitted to a Fermi function using those temperatures andan additional convoluted Gaussian to account for the finite resolution of the exper-imental setup.trade off between energy resolution and measurement time has to be made, keep-ing in mind factors like sample aging and availability of liquid helium.2.2 Scanning tunnelling microscopySince its first implementation [123], STM has developed into a powerful techniquethat gives access to the local topography on an atomic scale as well as spectroscopyof the occupied and unoccupied density of states near the Fermi level. It relies onquantum mechanical tunnelling effects between an atomically sharp tip and thesample. A schematic setup for a STM experiment can be seen in Fig. 2.8a. Piezoscanners allow the precise control of the tip in the lateral directions as well as in the24Figure 2.6: Angle-resolved photoemission on monolayer graphene. (a) Fermisurface measured at one of the corners of the Brillouin zone (BZ). The Fermisurface shows signs of trigonal warping. (b) Cut along the kx direction as indicatedin (a). The Dirac point is at about 450 meV binding energy. (c) Similar cut as in(b), but along the ky direction. The sample was held at a temperature of 7 K.Figure 2.7: Fermi edge of monolayer graphene. The ARPES data for one linearbranch of the Dirac cone was integrated in momentum perpendicular to the Fermisurface. Data was taken at 7 K. (a) Fermi edge measured with an analyzer passenergy (PE) of 1 eV. The edge was fitted to a Fermi function convolved with aGaussian. The resulting energy resolution is ∆= (0.9±0.2)meV. (b) Fermi edgemeasured with an analyzer pass energy (PE) of 2 eV. The edge was fitted to a Fermifunction convolved with a Gaussian. The resulting energy resolution is ∆= (2.1±0.3)meV. As expected, the energy resolution worsens with increasing pass energy.25Figure 2.8: Scanning tunneling microscopy setup and energy schematic. (a)A piezo scanner setup allows the precise positioning of the tip with respect to thesample. A bias voltage Vbias is applied between the sample and the tip. The result-ing tunneling current is amplified and measured in a feedback loop. (b) The samplewith work function φsample and the tip with work function φtip are at a distance dfrom each other. Due to the applied voltage, electrons can tunnel between tip andsample. Note that for useful spectroscopic information of the sample, the densityof states of the tip has to be flat in the bias voltage range in question.direction perpendicular to the sample. An applied bias voltage allows electrons toeither tunnel into or out of the sample. The resulting current (typically nA or pA)is amplified and measured in a feedback loop. To obtain spectroscopic informationabout the density of states, the bias voltage can be varied. This technique alwaysmeasures the convolution of the density of states of the sample with the densityof states with tip. Therefore a tip material with a flat density of states aroundthe Fermi level is mandated (see Fig. 2.8b). For the experiments shown in thisthesis, a platinum-iridium tip cut from a wire was used. Note that in general STMmeasures the momentum-integrated density of states of the sample, but for certainmaterials the band dispersion can be indirectly accessed by analyzing quasi-particleinterference patterns [124]. As a technique directly measuring the sample’s surface,STM is a UHV experiment. Due to the strong dependence of the tunnelling signal onthe tip’s lateral position and distance from the sample, STM is highly susceptible tomechanical noise and thus has to be properly isolated from any sources of vibration.Topographic images of the sample are usually taken in one of two modes ofoperation (see Fig. 2.9). In constant current mode, the height of the tip with respect26Figure 2.9: Scanning tunneling microscopy modes. (a) In constant currentmode, the height of the tip is adjusted during the scan to keep the tunneling currentconstant. (b) In constant height mode, the vertical position of the tip is fixed andthe tunneling current varies during the measurement.to the sample is adjusted as it is scanned across the sample to keep the tunnelingcurrent constant. In constant height mode, the vertical position of the tip is fixedduring the scan and the tunneling current varies. The constant height mode shouldbe used carefully and only on flat samples, as the danger of crashing the tip intothe sample is given.For a brief introduction into the theory of STM, it is suitable to start with asimple picture of electrons tunneling through a potential barrier [125–127]. Thiscauses a current that depends on the distance between the tip and the sample as wellas the applied bias voltage. For a more in-depth treatment, several more involvedtheories of STM have been developed in the literature [128–131]. In the case of anegative bias voltage being applied to the sample with respect to the tip, electronstend to flow from the sample to the tip. Using time dependent perturbation theory,the elastic current can be estimated by:Isample→tip =−2e 2pih¯ |M|2(ρs(ε) · f (ε))(ρt(ε+ eV ) · [1− f (ε+ eV )]). (2.12)Here ε is the energy with respect to the Fermi level of the sample, e is the electroncharge, the factor of 2 arises from the spin degeneracy, |M|2 is the matrix element,ρs and ρt are the density of states of the sample and tip, respectively, and f (ε) isthe Fermi function. Although most of the electrons will tunnel from the sample to27the tip in case of a negative bias voltage, some electrons will nevertheless tunnelfrom the tip to the sample. This contribution can be summarized as follows:Itip→sample =−2e 2pih¯ |M|2(ρt(ε+ eV ) · f (ε+ eV ))(ρs(ε) · [1− f (ε)]). (2.13)The net tunneling current is then the difference between those two contributionsafter integrating over all energies:I =−4pieh¯∫|M|2ρs(ε)ρt(ε+eV ){ f (ε)[1− f (ε+ eV )]− [1− f (ε)] f (ε+ eV )}dε.(2.14)This lengthy expression can simplified when keeping in mind that tunneling canonly take place from occupied to unoccupied states. Whether a state is occupiedor not is determined by the Fermi function, which at typical measurement tem-peratures of about 4.2 K (kBT = 0.36meV) can be well approximated by a stepfunction (1 for energies below the Fermi level and 0 for energies above the Fermilevel). Hence, we can approximate Eqn. 2.14 withI ≈−4pieh¯∫ 0−eV|M|2ρs(ε)ρt(ε+ eV )dε. (2.15)As a second simplification, we can make use of the fact that we required the tipmaterial to have a flat density of states around the Fermi level. Hence, the termρt(ε+ eV ) can be assumed constant over the integral range:I ≈−4pieh¯ ρt(0)∫ 0−eV|M|2ρs(ε)dε. (2.16)Next, we make the assumption that the matrix element can also be treated as a con-stant. This is only true for limited energy ranges, when the exponentially decayingwave functions of the sample and the tip are independent and do not influence eachother significantly [124, 125, 132]:I ≈−4pieh¯ ρt(0)|M|2∫ 0−eVρs(ε)dε. (2.17)For an estimate of the matrix element, we can turn to our simple one-dimensionalmodel of a wave function tunneling through a square potential barrier. In this case28Figure 2.10: STM tip conditioning. Topography images of terpyridine-phenyl-terpyridine (TPT) molecules on a Ag(111) surface. (a) Image taken with a “bad”tip with Vbias = +500mV and Itun. = 10pA. The molecules look slightly blurryand replicas in the vertical direction are visible. This is indicative of a not per-fectly sharp tip and more than one active tunneling junction. The Ag(111) sur-face shows the interference pattern of electron waves in the substrate scatteringoff the molecules. (b) Image taken with a “good” tip with Vbias = +7mV andItun. = 330pA. Molecules appear sharper and no replicas are visible. Note the dif-ferent wavelength of the interference pattern due to the changed bias voltage. Bothimages were taken at a temperature of 4.2 K.the Wentzel-Kramers-Brillouin (WKB) approximation yields |M|2 = exp(−2γ) forthe tunneling probability withγ = dh¯√2mβ . (2.18)Here d is the distance between sample and tip (width of the barrier) and β dependson the work functions of the tip and the sample (height of the barrier). This ex-ponential relationship between the tunnelling current and the distance ensures theextremely high spatial resolution of STM experiments. It also means that on flatsamples the macroscopic shape of the tip is rather unimportant, as tunneling pri-marily happens from the outermost atom. The actual distance from the tip to thesample is not known and can change either due to the topography or due to a changein the density of states (defects, impurities, etc.). Calculations suggest a distanceof several a˚ngstro¨ms to be in a regime where vacuum tunnelling dominates overpoint contact phenomena [133].29From Eqn. 2.17 we see that STM measures the integrated density of states ofthe sample for the range of accessible bias voltages. For negative bias voltages,electrons tunnel from the sample to the tip and the occupied density of states ofthe sample are measured. For positive bias voltages, electrons tunnel from the tipto the sample and the unoccupied density of states of the sample are probed. Toaccess the density of states, we have to look at the normalized derivative of the biasvoltage dependent tunnelling current [134].DOS =dI/dV√C2+(I/V )2(2.19)A small constant C is often added to avoid the singularity around 0 V. It shouldbe noted that, just like in ARPES, STM does not measure the ground state, but anexcited state as electrons are either added or removed from the material. Hence,effects like charging, Coulomb interactions, or screening can play an importantrole [135–137].As we have learned, a well-conditioned tip is essential for STM measurements.This includes the structure (ideally sharp with a single atom at the tip) and the elec-tronic properties (flat density of states around the Fermi energy). To accomplishthis, a calibration sample with known structures and electronic properties is needed.In our case we use a Ag(111) surface which has been cleaned by cycles of ion sput-tering and annealing. Next, some terpyridine-phenyl-terpyridine (TPT) moleculeswere deposited on it. Scanning over a region with molecules should yield sharpstructures with no additional ghost replicas in any direction (compare Fig. 2.10).Replicas are an indication that the tip has multiple active tunnelling atoms thatscan an area one after another. There are several ways to condition the tip in caseimprovements to the tip structure are deemed necessary: (i) for large changes, thetip can be rammed into the surface (typically several nanometers deep) and draggedaround , (ii) for smaller changes, the tip can be poked into the surface (typicallyless than one nanometer deep) and the resulting protrusion can be scanned to get anidea of the tip geometry, and (iii) voltage pulses can be applied to the tip to removematerial or adsorbates. In reality, the preparation of a “good” tip often requires acombination of all methods as well as an experienced eye, patience, and a bit ofluck.30Figure 2.11: Ag(111) surface state with STM. (a) I(V ) curve on a Ag(111) sur-face with a “bad” tip. In addition to the kink at -60 meV corresponding to thesurface state, several additional features due to tip states are visible. (b) Normal-ized dI/dV curve of (a). The density of states is not flat, indicating a “bad” tip. (c)I(V ) curve on a Ag(111) surface with a “good” tip. The surface state at -60 meVis the only clear feature. (d) Normalized dI/dV curve of (c). Apart from the ex-pected step in the density of states due to the surface state, the tip does not induceany pronounced additional features.31Figure 2.12: STM on epitaxially grown monolayer graphene on SiC. (a)Overview topography image taken at Vbias = 30mV and Itun. = 2pA. In additionto the honeycomb lattice of graphene, the (6√3× 6√3)R30◦ superlattice due tothe lattice mismatch of graphene and the SiC substrate is visible. (b) Close up to-pography image taken at Vbias = 30mV and Itun. = 2pA. The individual rings ofthe carbon lattice are clearly visible. The variation in tunneling current is probablyinduced by electronic or structural inhomogeneities in the underlying substrate orslight buckling of the graphene surface [140].Besides the geometry of the tip, also its electronic properties have to be checked.The Ag(111) surface shows a well-known and pronounced surface state with aparabolic dispersion opening at -60 meV binding energy [138, 139]. This two-dimensional electron gas at the surface leads to a step in the density of states.Other than that, the density of states of silver is expected to be flat around theFermi energy. In the I(V ) curve, the surface state should show up as a kink with alinear dependence on either side, which translates to a step function in the densityof states after differentiation (compare Fig. 2.11). Any additional features indicatethat the density of states of the tip is not flat. In that case, the same tip conditioningprocedures as for the tip geometry can be applied. In general, a structurally “good”tip does not imply an electronically “good” tip or vice versa. Hence, after any tipalterations, both features should be checked.After the tip is presumed suitable for experiments, the calibration sample can beswitched with the actual monolayer graphene on SiC sample. A topographic over-view image and a close-up view are shown in Fig. 2.12. They show the expectedhoneycomb lattice of graphene as well as the well-known (6√3× 6√3)R30◦ su-32per structure. The latter arises due to the lattice mismatch of the graphene andthe underlying SiC substrate. Note here that tunneling spectroscopy with STM ongraphene has the additional complication that direct tunneling between the tip andthe graphene is not possible. This is due to the fact that the electrons in graphenereside near the K points at the corners of the BZ in momentum space and can notelastically tunnel. Inelastic tunneling is possible when taking for example phononsor substrate interactions into account [141–143].2.3 Resonant energy-integrated X-ray scatteringREXS is a powerful technique based on the scattering of photons (photon-in andphoton-out). It is sensitive to the charge, orbital, spin, and lattice degrees of free-dom, which has made REXS an increasingly popular choice for the study of in-tertwined ordering phenomena in a range of quantum materials [144]. The reso-nant nature of the technique allows the detection of weak and short-range phases.Among the great successes of REXS were the detection and characterization of thebroken-symmetry charge density order in the cuprate family of high-temperaturesuperconductors [145–153] and the orbital ordering in the manganites [154–156].In the following section, a brief introduction into the theoretical background ofREXS is given. For a more comprehensive review, several publications are avail-able [157–161].The interaction of an electromagnetic field with a solid can be described by aneffective nonrelativistic coupling Hamiltonian Htot :Htot =∑j{12m[p j−ecA(r j, t)]2+V (r j, t)}+∑j 6=ke2|r j− rk|2 +HEM= Hel +HEM︸ ︷︷ ︸H0+emc∑jA(r j, t) ·p j︸ ︷︷ ︸H linint+e22mc2∑jA2(r j, t)︸ ︷︷ ︸Hquadint.(2.20)Here m is the fundamental mass of an electron, e is the charge of an electron, p jis the momentum of the jth electron, r j is the position of the j-th electron, A(r j, t)represents the vector potential, V (r, t) is the lattice potential, and the term e2|r−r′|2is the Coulomb interaction term. Thus the noninteracting Hamiltonian H0 can be33split into an electronic part Hel and a part for the electromagnetic field HEM =∑q,ν h¯ω[a†ν(q)aν(q)+1/2] describing photons with wave vector q, polarization ν ,and energy h¯ω being either created (a†ν ) or annihilated (aν ). The interaction part ofthe Hamiltonian can be split into a part linear in the vector potential H linint and a partquadratic in the vector potential Hquadint . In general, we are interested in calculatingthe probability for a transition from an initial state |Ψ〉i = |ψGS〉el × |φi〉EM to afinal state |Ψ〉 f = |ψGS〉el ×|φ f 〉EM with the photon states |φi〉EM and |φ f 〉EM andthe electronic part |ψGS〉el , which we assume to be in the ground state. To calculatethis transition probability ωi→ f , we can turn to the generalized Fermi’s golden rule[162]:ωi→ f = 2pi|〈Ψi|T |Ψ f 〉|2δ (E f −Ei). (2.21)Here the delta function guarantees the conservation of energy and the transfer ma-trix is defined as follows:T = Hint +Hint1Ei−H0+ iηHint +Hint1Ei−H0+ iηHint1Ei−H0+ iηHint + ....(2.22)Before proceeding, we should note that for photon scattering techniques like REXS,we are looking for operator combinations in the interaction which annihilate a pho-ton with wave vector q and create a photon with wave vector q’ (a(q)a†(q′)).Hence, interaction terms in the transfer matrix which are quadratic in the vec-tor potential are required, as A(r, t) ∝ ∑q,ν ε ν · [exp(iq · r− iωt)a†ν(q)+ h.c.] (ε νis the polarization vector of polarization state ν). If we now look at Eqn. 2.20and Eqn. 2.22, we see that we get two contributions. The first one combines thequadratic interaction operator Hquadint with the first term in Eqn. 2.22 and the secondone combines the linear interaction operator H linint with the second term in Eqn. 2.22.ω(1)i→ f = 2pi∣∣∣ e22mc2〈Ψi|∑jA2(r j, t) |Ψ f 〉∣∣∣2 (2.23)ω(2)i→ f = 2pi∣∣∣( emc)2∑M〈Ψi|∑ j A(r j, t) ·p j |ΨM〉〈ΨM|∑k A(rk, t) ·pk |Ψ f 〉Ei−EM + iΓM∣∣∣2 (2.24)Here the delta functions from Eqn. 2.21 are dropped for brevity and |ΨM〉 describesa generic excited state of the solid interacting with the electromagnetic field with34energy EM and lifetime h¯/ΓM. If we expand the square of the vector potential underthe assumption of an elastic scattering process (ωin =ωout) A2(r, t)∝ (ε νin ,ε νout )×exp(i(qout−qin) ·r) ·a†νout (qout)aνin(qin), use the definitions of the initial state |Ψi〉and final state |Ψ f 〉 from above, use the fact that Ei = EGS +[nqin h¯ωqin +1/2] andEM = Em + [(nqin − 1)h¯ωqin + 1/2], and consider that a†νout (qout)aνin(qin) |φi〉EM ∝|φ f 〉EM, we can rewrite Eqn. 2.23 and Eqn. 2.24 [144].ω(1)i→ f = |〈ψGS|∑jexp(−iQ · r j) |ψGS〉|2 ∝ |〈ψGS|ρ(Q) |ψGS〉|2 (2.25)ω(2)i→ f =∣∣∣∑m∑j,k〈ψGS|ε νout p jexp(iqout · r j) |ψm〉〈ψm|ε νinpkexp(−iqin · rk) |ψGS〉EGS−Em+ h¯ω+ iΓm∣∣∣2(2.26)Here ρ(Q) is the Fourier transform of the electron density operator ρ(r)=∑ j δ (r−r j) and Q = qin−qout is the momentum transfer between the scattered photon andthe sample. It should be emphasized that we have gone from a description of theintermediate state including both the electronic and photonic part of the wave func-tion |ΨM〉 to a description of the intermediate state only involving the electronicpart of the wave function |Ψm〉 here. |Ψm〉 is in general still an excited many-bodystate with a core hole. The first process in Eqn. 2.25 does not involve an interme-diate state and is directly proportional to the square of the electronic density in theground state. This leads to the conventional nonresonant X-ray diffraction (XRD)signal. Because it relies on the total number of electrons, it is typically moresensitive to atoms with higher atomic number and core electrons as they usuallyoutnumber the valence electrons significantly. The second process in Eqn. 2.26 isassociated with REXS. It can be understood more intuitively as a two-step process:First, an incoming photon promotes an electron into an excited state leaving a corehole behind. Second, a scattered photon is re-emitted and the core hole is filledagain, leaving the sample in its ground state. The two processes are schematicallyillustrated in Fig. 2.13a. The advantage of REXS is now the resonant enhancementof the signal close to an absorption edge (see Fig. 2.13b). While the XRD signal ismostly independent of the photon energy, the REXS is peaked around the resonanceand decays to zero away from it. By picking a specific absorption edge via tuningthe photon energy, one can select certain chemical elements and orbitals. Further,35Figure 2.13: Schematic of resonant X-ray scattering process. (a) Comparisonbetween nonresonant (left) and resonant (right) scattering process. The resonantscattering process can be described by an intermediate state |Ψm〉 with a core holeconnecting the ground states |ΨGS〉. (b) Effect of resonance enhancement close toan absorption edge. The nonresonant measurement (dashed blue) is mostly inde-pendent of the photon energy hν , while the resonant measurement (red) shows apeak of width Γli which decays to zero away from the resonance. Figure adaptedfrom [144].the signal is sensitive to spin excitations through the spin-orbit interaction of thecore hole in the intermediate state. Experimentally, signal enhancements of more> 103 have been found [163, 164], allowing the detection of weak and short-rangeordering phenomena above the noise level, even if they are not accompanied by alattice distortion.If we now look at an actual realization of an REXS experiment, we first notethat the technique requires a photon source with tunable photon energy in the X-ray regime. This means the use of synchrotron facilities is necessary. A numberof beamlines dedicated to resonant X-ray scattering experiments are already avail-able or under construction worldwide (ALS, APS, BESSY, CLS, DESY, Diamond,ESRF, NSLS-II, NSRRC, SLS, SOLEIL, Spring-8, SSRL) [144]. They can in gen-eral be divided into hard X-ray (>2.5 keV photon energy) and soft X-ray facilities(up to 2 keV photon energy). For the purposes of this thesis, we are looking atedges in the soft X-ray regime, which are available at the REXS endstation at theCanadian Light Source (CLS) [165]. In contrast to hard X-rays, soft X-rays addthe additional complications of covering a smaller part of momentum space dueto the smaller momentum of the photons and requiring the entire experiment to36Figure 2.14: Geometry of resonant X-ray scattering experiment. (a) The di-rection of the incoming photon (qin,hνin) is fixed to the beamline direction andthe outgoing scattered photon (qout ,hνout ≈ hνin) is detected. The sample can berotated around various axes (θ , χ , and ϕ) as well as moved laterally (x, y, and z).The direction of the outgoing photon is determined by the scattering angle θsc. (b)Geometry showing the magnitude of the exchanged momentum Q between pho-ton and sample. By projecting Q onto the sample surface, the parallel (Q‖) andperpendicular components (Q⊥) of the transferred momentum can be calculated.Adapted from [144].be in UHV due to the higher attenuation of photons with smaller photon energy.An illustration of the experimental setup is shown in Fig. 2.14. The direction ofthe incoming photon is fixed by the beamline, and the sample (θ ) and the detector(2θ = θsc) can be rotated individually. The sample can additionally be rotated inχ and ϕ in a limited range as well as moved in all lateral directions x, y, and zfor sample alignment. The sample stage can further be cooled with liquid heliumand is equipped with heaters, allowing a continuous adjustment of the tempera-ture. In our case, the photon detectors do not discriminate photon polarization orphoton energy, so that the signal can be viewed as an energy integrated spectrumcomprised of the elastic and inelastic part. In general, experimental schemes dif-ferentiating different energies of the scattered light are available (resonant inelasticX-ray scattering (RIXS)), but come at the cost of a reduced efficiency. Under theassumption of a nearly elastic scattering process (hνout ≈ hνin), the magnitude of37the transferred momentum can be expressed as Q = 2qin× sin(θsc/2) [144]. Whenthe vector Q is projected onto the plane defining the sample surface, the parallelcomponent Q‖ and perpendicular component Q⊥ of the transferred momentum canbe extracted (see Fig. 2.14b).2.4 Low energy electron diffractionLEED is a useful technique for the characterization of surfaces based on the diffrac-tion of electrons off the sample [166–169]. It relies on the groundbreaking hy-pothesis by Louis de Broglie that all particles can be associated with a wave likecharacter [170]:λ =h√2mEkin. (2.27)Here the wavelength λ depends on the mass of the electron m and the kineticenergy Ekin. h is the Planck constant. To obtain constructive interference betweenan incoming electron and a scattered electron the Laue condition has to be fulfilled[171]:k−k0 = ha∗+ kb∗+ lc∗ = Ghkl. (2.28)Here k0 is the wave vector of the incident electron, k is the wave vector of thediffracted electron, a∗, b∗ and c∗ are reciprocal lattice vectors, and hkl are a set ofinteger numbers. Note that the LEED process is elastic and therefore the magnitudeof the electron wave vector remains unchanged (|k|= |k0|). For a two-dimensionalmaterial like graphene, the relationship in Eqn. 2.28 reduces to [168]:k‖−k‖0 = ha∗+ kb∗ = Ghk. (2.29)Even though the first electron diffraction experiments were successfully performedearly on [172, 173], it took until the 1960s to establish LEED as a tool in the anal-ysis of surfaces [174–177]. This is mainly due to fact that LEED is a very surface-sensitive technique. Hence, atomically clean and flat surfaces are necessary, typi-cally requiring UHV conditions and sample preparation. The surface sensitivity isdirectly related to the electron inelastic mean free path in a solid (compare withARPES in Fig. 2.2) [107]. In the typical energy range (10 eV to 100 eV) the pen-38Figure 2.15: Schematic setup of a low energy electron setup. A monochromaticelectron beam from an electron gun is collimated by a Wehnelt cap. The diffractedelectrons pass through the suppressor grids before hitting a phosphor screen fordetection. The sample is grounded.etration depth is on the order of several A˚ngstroms, corresponding to a few unitcells of most materials. In general, LEED experiments can be performed in one oftwo ways: (i) qualitative analysis of the position and “sharpness” of the diffractionspots, and (ii) recording the intensities and profiles of individual spots as a functionof the electron energy. The former is useful to determine the orientation, symmetryand general quality of a sample (we will stick to this approach in this thesis). Thelatter can give additional information about atomic positions, defects, terrace sizes,etc. on the surface, but requires a more comprehensive data analysis and theoreticalmodelling [178, 179].A schematic illustration of a LEED experiment is shown in Fig. 2.15. An elec-tron gun produces a monochromatic beam of electrons, which is collimated by aWehnelt cap before hitting the electrically grounded sample. The scattered elec-trons pass through several suppressor grids and are then detected on a phosphorscreen. The suppressor grids are supposed to repel inelastically scattered electronswith lower kinetic energy. Typical spot sizes of the electron beam on the sampleare between 0.1 mm and 0.5 mm depending on the settings of the electron optics.Before looking at an actual experimental LEED image of monolayer grapheneon SiC, we can use the structural information available for the sample system tosimulate the expected LEED pattern (see Fig. 2.16). The hexagonal structure of39Figure 2.16: Simulation of the LEED pattern for graphene on SiC. The hexag-onal diffraction patterns of graphene (red) and the SiC substrate (blue) are rotatedagainst each other by 30◦. SiC has a slightly larger lattice constant than graphenein real space, leading to slightly smaller reciprocal lattice vectors in the LEED im-age. The lattice mismatch also leads to a (6√3×6√3)R30◦ superstructure (green).Note that not all diffraction spots for the superstructure are shown for simplicityand easier comparison with the experimental LEED pattern (see Fig. 2.17).graphene and the SiC substrate are rotated by 30◦ with respect to each other. TheSiC has a slightly larger lattice constant than graphene in real space. In reciprocalspace probed by LEED, the situation is reversed, so we expect the SiC diffractionspots to be slightly closer to the center of the image in comparison to the graphenediffraction spots. In addition, we have to keep the (6√3× 6√3)R30◦ superlatticebetween the graphene and the substrate in mind, which arises from the lattice mis-match. The comparatively large unit cell of the superstructure in real space leadsto short reciprocal scattering vectors in the LEED image rotated by 30◦ form thegraphene spots. Note that in Fig. 2.16 not all spots of the superstructure in thevisible area are shown for simplicity and better comparison to the experimentaldata. Indeed the experimental LEED pattern of monolayer graphene on SiC showsthe expected features (see Fig. 2.17). The diffraction spots of the SiC substrate40Figure 2.17: LEED pattern for monolayer graphene on SiC. Experimental LEEDpattern obtained at 66 eV electron energy with the sample held at 6 K. The diffrac-tion spots of the graphene are marked in red, the diffraction spots of the SiC sub-strate in blue, and the diffraction spots corresponding to the reconstructed superlat-tice between sample and substrate in green. The dark feature visible in the centerof the image is due to the electron gun and electrical connections leading to it.are considerably weaker compared to the graphene spots, as the former is buriedbeneath the graphene. Also note that only a limited number of superlattice spotsare visible, in particular around the graphene spots. This could be due to additionalinterference effects and can change as the electron energy is modulated and higher-order diffraction spots come into the range of the detector. For the purposes of thisthesis, we try to align the graphene sample in the depicted fashion in Fig. 2.17, toallow easy access to one of the corners of the BZ with the given photon energy inARPES.2.5 Raman spectroscopyRaman is a spectroscopic method used to study low-energy excitations in materi-als [180–182]. It is commonly applied in chemistry to identify molecules through41vibrational fingerprints or in solid state physics to study phonons or other collec-tive excitations of a crystal [183–185]. Raman is based on the inelastic scatteringof monochromatic light, which was first theorized in 1923 [186]. The effect wasexperimentally realized shortly after in 1928 by C. V. Raman and independently byG. Landsberg and L. Mandelstam [187–189]. The process of Raman spectroscopyis illustrated in Fig. 2.18. Starting from the electronic ground state, an incomingphoton excites the material into a virtual state. Most of the time the decay returnsit to the same vibrational level (Rayleigh scattering), meaning the emitted pho-ton has the exact same energy as the incoming photon. Nevertheless, for a smallnumber of photons the energy changes, as a vibrational mode in the material iseither excited or absorbed by the photon. This is the Raman effect. If a vibrationalmode in the material is excited, the energy of the outgoing photon is reduced bythe energy of that mode (Stokes scattering). If a vibrational mode instead is de-excited, the emitted photon has an increased energy (anti-Stokes scattering). Theratio between Stokes and anti-Stokes scattering depends on the energy of the vibra-tional modes and the temperature. For typical phonon energies and temperatures,materials are mostly in the electronic and vibrational ground state, so that Stokesscattering dominates. Note that for known material parameters, the ratio betweenStokes and anti-Stokes scattering can be used to determine the temperature of thesample [190–192].The theory of Raman scattering can be illustrated by a simple model based onthe polarizability of a material. For a more in-depth review a number of publi-cations are available [193–195]. Excitations in a material (e.g. phonons) can bedescribed by a periodic motion:q = q0cos(2piνvibt). (2.30)Here q is the displacement, q0 the amplitude of the oscillation, and νvib is the char-acteristic frequency of the oscillation. The electromagnetic field of an incomingphoton induces a dipole moment P depending on the polarizability α of the mate-rial:P = αE0cos(2piν0t). (2.31)42Figure 2.18: Process of Raman spectroscopy. (a) Both the electronic groundstate (GS) and the excited state (ES) have a series of vibrational states associatedwith them. Typically the vibrations have a smaller energy scale then the differencebetween the electronic GS and ES. An incoming photon connects to a virtual statein the band gap. The emission of the scattered photon can either happen to thesame vibrational state (Rayleigh scattering – black), to a state with higher vibra-tional energy (Stokes scattering – red), or to a state with lower vibrational energy(Anti-Stokes scattering – blue). (b) Energy shifts for the three different scatter-ing processes. For Rayleigh scattering (black) no change in the photon energy isobserved. For Stokes scattering (red) the energy of the emitted photon is lower,and for anti-Stokes scattering (blue) the energy of the emitted photon is higher.The elastically scattered component is much stronger than the two inelastic com-ponents. The intensity ratio between the Stokes and anti-Stokes peaks depends onthe energies of the vibrations involved and the temperature.Here E0 is the amplitude of the electromagnetic field and ν0 is the energy-dependentfrequency of the photon. For small amplitudes the polarizability can be expandedin terms of the displacement:α = α0+q( ∂α∂ t )q=0+ .... (2.32)43Figure 2.19: Setup of a Raman spectroscopy experiment. Monochromatic lightfrom a laser is guided onto the sample. Optics of a microscope can be used toreduce the spot size of the beam on the sample and add lateral resolution to the ex-periment. The reflected light is analyzed according to its energy in a spectrometerwith a grating and then detected. Adapted from [196].Combining Eqn. 2.30, Eqn. 2.31, and Eqn. 2.32, we get a new expression for theinduced dipole moment in the material:P = α0E0cos(2piν0t)+q0cos(2piνvibt)E0cos(2piν0t)( ∂α∂ t )q=0. (2.33)The first term in Eqn. 2.33 describes the usual Rayleigh scattering process. Thesecond term is the basis of Raman scattering. It can be slightly rewritten using oneof the identities for trigonometric functions:12 q0E0(∂α∂ t )q=0[cos(2pi{ν0−νvib}t)+ cos(2pi{ν0+νvib}t)]. (2.34)Here the first term describes a dipole with a decreased frequency (Stokes scattering)and the second term a dipole with an increased frequency (anti-Stokes scattering).A typical experimental setup for Raman scattering is depicted in Fig. 2.19. As asource of intense monochromatic light, a laser is used. The light is guided through44Figure 2.20: Resonant Raman process in graphene. (a) Origin of the so calledRaman G peak of graphene. The absorption of a photon generates an electron-holepair. The recombination involves a Γ-point phonon with zero momentum. Theenergy difference between incoming and outgoing photon equals the energy of thephonon. (b) Origin of the so called Raman 2D peak of graphene. In addition tothe resonant creation of the electron-hole pair, the process involves two K-pointphonons that resonantly couple the K and K’ valleys. In this case the energy dif-ference between the incoming and outgoing photon equals the sum of the energiesof the two phonons involved. In both (a) and (b) optical transitions are depicted asblue arrows and phonon transitions are depicted as orange arrows. Adapted from[197].an optical setup onto the sample. Often a microscope is used to focus the light andreduce the spot size of the beam on the sample (≈ µm depending on the wavelengthof the used light). This allows us to scan the sample with the photon beam andadd lateral resolution to the experiment. The reflected light is guided through aspectrometer with a grating to discriminate different energies, then detected forread out.Raman spectroscopy has been and still is a powerful tool in the field of gra-phene. The linear dispersing bands forming cones around the Dirac points allowfor direct optical transitions and for the resonant creation of electron-hole pairsin a range of photon energies. The two dominant Raman features for monolayergraphene on SiC are the so called G and 2D peaks. The underlying processesfor both are illustrated in Fig. 2.20. The G peak process involves the creation ofan electron-hole pair through photon absorption and a single Γ-point phonon withzero momentum. The 2D peak process involves the creation of an electron-holepair through photon absorption and two K-point phonons which resonantly couple45Figure 2.21: Raman spectroscopy of monolayer graphene on SiC. Raman spec-trum taken with a helium-neon laser (632.8 nm) with the sample at room temper-ature. (a) The low-energy part of the spectrum is dominated by sharp transitionscorresponding to the SiC substrate. (b) At higher wave numbers, additional SiCpeaks overlap with the graphene G peak (E2g stretching phonon mode at the Γpoint). The G peak is indicated with a blue star slightly below 1600 cm−1. Theso-called graphene 2D peak appears due to an intervalley process involving twophonons. It can be used to determine the number of graphene layers and interac-tions with a substrate. (c) Close-up of the graphene 2D peak. Note the differentintensity scales from (a) – (c).46the two valleys K and K′ in graphene. In the first process the energy differencebetween the incoming and outgoing photon is equal to the energy of the singlephonon, while in the second process the energy difference is equal to the sum of theenergies of the two phonons involved [198–202]. In both cases the total momen-tum and energy of all particles must be conserved. At typical photon energies usedfor Raman experiments on graphene the momenta of the photons can be neglectedand the optical transitions can be depicted as vertical transitions in the electronicband structure (compare Fig. 2.21). Raman spectroscopy can be used as a tool todetermine the number of layers of graphene, analyze sample quality (defects, flakesize, etc.), and to study low-energy excitations like phonons and their coupling toother degrees of freedom [197, 202–206]. A typical Raman spectrum of monolayergraphene epitaxially grown on a SiC substrate is shown in Fig. 2.21. The data wastaken with light from a helium-neon laser (632.8 nm) and the sample at room tem-perature. At low energies, the spectrum is dominated by intense peaks correspond-ing to excitations in the SiC substrate. Around 1600 cm−1 the graphene G peak isvisible, but also overlaps with signal from the substrate. By comparing our datawith available Raman spectra for monolayer graphene on SiC form the literaturewe can identify the G peak (blue star in Fig. 2.21b) [207–210]. It arises from theE2g stretching phonon mode at the Γ point [203]. Around 2650 cm−1, the so-calledgraphene 2D peak originates from an intervalley scattering process involving twophonon modes [203]. It is sometimes also called G∗ peak. The position and widthof the 2D feature can be used to identify different numbers of graphene layers andinteractions with different substrates [207, 211, 212]. Especially noteworthy hereis the work by Lee et al. [213] which showed that for graphene on SiC the widthof the Raman 2D peak is the fingerprint to differentiate between different num-bers of graphene layers. They also showed that Raman features shift if grapheneis transferred from SiC to another substrate, while the width of features remainsunchanged and is intrinsic to the graphene. Finally, Mueller et al. [214] showedthat by combining the analysis of the graphene G and 2D peaks it is possible todisentangle the effects from for example strain and doping in graphene. Unfortu-nately, this is not possible in our data due to the overlap of the G peak with featuresfrom the underlying SiC substrate.47Chapter 3Strain-induced Landau levels ingrapheneIn the presence of strong magnetic fields, two-dimensional (2D) electron systemsdisplay highly degenerate quantized energy levels called Landau levels (LLs) [27].When the Fermi energy is placed within the energy gap between these LLs, thesystem bulk is insulating and charge current is carried by gapless edge modes.This is the quantum Hall effect, belonging to the remarkable class of macroscopicquantum phenomena [215–218] and the first member of an ever-growing family oftopological states [219, 220]. While angle-resolved photoemission spectroscopy(ARPES) has been a powerful tool to investigate numerous quantum phases of mat-ter [84, 221, 222], the traditional quantum Hall states – and thus their momentum-resolved structure – have remained inaccessible. Such observations are hinderedby the fact that ARPES measurements are incompatible with the application ofmagnetic fields. Here, we circumvent this by using graphene’s [1, 2, 17] peculiarproperty of exhibiting large pseudomagnetic fields under particular strain patterns[85] to visualize the momentum-space structure of electrons in the pseudo-quantumHall regime. By measuring the unique energy spacing of the resulting pseudo-LLswith ARPES, we confirm the Dirac nature of the electrons in graphene and extracta pseudomagnetic field strength of B = 41 T. This momentum-resolved study ofthe quantum Hall phase up to room temperature is made possible by exploitingshallow triangular nanoprisms in the SiC substrates that generate large, uniform48Figure 3.1: Identification of nanoprisms. (a) Horizontal derivative AFM topog-raphy image of our monolayer graphene grown on a SiC substrate. Triangularnanoprisms are dispersed on the surface. Inset: AFM topography image of thesame area. Substrate terrace steps are about 10nm in height. (b) Top: Close-upAFM topography of the area indicated by the black box in (a). Bottom: Line cutthrough the AFM data marked by the purple line in the close-up. (c) Overview STMtopography image (200 nm × 200 nm, Vsample = 100mV, Itun. = 2pA) showing asingle nanoprism.pseudomagnetic fields, arising from strain, confirmed by scanning tunnelling mi-croscopy (STM) and model calculations. Our work demonstrates the feasibility ofexploiting strain-induced quantum phases in 2D Dirac materials on a wafer-scalesize, opening the field to a range of new applications.Graphene was the first material in which a member of the striking class ofmacroscopic quantum phenomena [215–217, 223] – the quantum Hall effect (QHE)[27] – could be observed at room temperature, when subject to large magnetic fields[37]. In the quantum Hall state, charge carriers are forced into cyclotron orbitswith quantized radii and energies known as LLs when subjected to the influence ofa magnetic field. In order to observe this effect, certain conditions must be met:the magnetic field must be large enough that the resulting spacing between LLs islarger than the thermal energy (∆ELL > kBT ); the charge carrier lifetime betweenscattering events must be longer than the characteristic time of the cyclotron orbit(tlife > 1/ωc); and the magnetic field must be uniform on length scales greater thanthe LL orbit. This typically mandates the need for cryogenic temperatures, cleanmaterials, and large applied magnetic fields. Dirac fermions in graphene provide away to lift these restrictions: Under certain strain patterns, graphene’s electrons be-49have as if they were under the influence of large magnetic fields, without applyingan actual field from outside the material [85, 86, 224, 225]. These so-called pseu-domagnetic fields only couple to the relativistic electrons around the Dirac pointand, under the QHE conditions above, lead to the formation of flat, quantized LLs.This has been successfully observed using a range of methods [86, 224, 225], butwas so far restricted to small regions, which severely limits its applicability.Here, we directly visualize the formation of flat LLs close to the Fermi energyinduced by pseudomagnetic fields on wafer-scale semiconductor samples. By mea-suring the hallmark√n energy spacing and momentum dependence of the ensuingpseudo-LLs with angle-resolved photoemission spectroscopy (ARPES), and withthe aid of model calculations, we confirm their quantum Hall nature and extract apseudomagnetic field strength of B = 41 T. This is made possible by the presence ofa distribution of triangular nanoprisms underneath the monolayer graphene in oursamples based on the well-established platform of epitaxial graphene on SiC sub-strates [57, 61, 64, 226], as revealed by a combination of atomic force microscopy(AFM) and scanning tunneling microscopy (STM) measurements. STM experimentswere performed at UBC under ultra-high vacuum conditions (< 5× 10−11 mbar)using a low-temperature scanning tunnelling microscope (Scienta Omicron) at liq-uid helium temperatures (∼ 4.2 K). All images were acquired in constant-currentmode using a cut platinum-iridium tip, which was conditioned by voltage pulsingand gentle indentation into a Ag(111) crystal. The samples were annealed at 550◦Covernight with a final pressure of p = 3×10−10 mbar in situ prior to the STM mea-surements. Graphene samples with a carbon buffer layer were epitaxially grownon commercial 6H-SiC substrates. The substrates were hydrogen-etched prior tothe growth under argon atmosphere. Details are described by S. Forti and U. Starke[227]. AFM characterisation measurements were taken at the Max Planck Institutein Stuttgart.Our topographic images of these samples (Fig. 3.1a inset) exhibit the well-known terraces and step edges of graphene grown on 6H-SiC [57], which are due toa miscut of the wafers from the (0001) direction of up to 0.1◦. A population of tri-angular nanoscale features are identified on the terraces of our samples (Fig. 3.1a),which are like those reported on similar substrates [228, 229]. These nanoprismsappear during the growth process of graphene on 6H-SiC and are controllable by50Figure 3.2: AFM height distribution. Height distribution for the AFM image inFig. 3.1b (Top). Two Gaussians (red) can be fitted to the data to extract the depthsof the nanoprisms. The integrated fraction curve is shown in yellow.Figure 3.3: Graphene layer coverage (a) STM image taken across the edge of ananoprism (Vsample = 30mV, Itun. = 10pA). The graphene grows smoothly overthe step without interruption. (b) AFM adhesion image taken in the same region asshown in Fig. 3.1a. The image shows no contrast between the nanoprisms and thesurrounding terraces (black box), thus clearly indicating that the nanoprisms arecovered by monolayer graphene.51Figure 3.4: Substrate-induced strain. (a) Schematic structure of 6H-SiC, show-ing its layered ABCACB stacking order with epitaxial graphene on top (yellow).Inside the nanoprism, a single layer within the unit cell is missing, exposing thegraphene to a different substrate surface termination, as illustrated in the top view.The carbon buffer layer is not shown for clarity. (b) Atomically resolved STMimages (10 nm × 10 nm, Vsample = 30mV, Itun. = 2pA) inside (top) and outside(bottom) of the nanoprism. (c) Difference map of the two Fourier transformed(FT) images in (b) visualizing the strain pattern inside the nanoprism.the argon flow in the chamber [229]. They cover between 5% and 10% of theterraces, which is supported by looking at the height distribution of the pixels inthe AFM image in Fig. 3.1b (top). We can determine the depths of the nanoprismsas well as estimate the coverage of the nanoprisms on the sample (see Fig. 3.2).The difference in the position of the two fitted Gaussians leads to a depth of thenanoprisms of (2.7±0.7) A˚. The integrated fraction curve indicates that about 5%to 10% of the total area is covered with nanoprisms. Further, the nanoprisms arecompletely covered by monolayer graphene, which is being demonstrated by ourAFM adhesion images (see 3.3b). Adhesion images correspond to the force neces-sary to retract the tip from the sample. Adhesion is sensitive to the graphene cov-erage on the sample and can thus distinguish between zero-layer, monolayer, andbilayer graphene with sensitivity to grain boundaries [230, 231]. The AFM adhe-sion image in Fig. 3.3b (taken in the same region as in Fig. 3.1a) shows no contrastbetween the nanoprisms and the surrounding terraces, thus clearly indicating thatthe nanoprisms are covered by monolayer graphene.52The nanoprisms are equilateral triangles oriented in the same direction with anarrow size distribution around 300nm side length. They are about (2.7± 0.7) A˚deep (Fig. 3.1b), which corresponds to a single missing SiC double layer or 16 of the6H-SiC unit cell. This leads to a change in the registry between the silicon atomsin the top layer of the substrate and the graphene as illustrated in Fig. 3.4d. Thestrain created inside the nanoprisms cannot be relieved, because the nanostructuresare continuously covered by monolayer graphene without additional grain bound-aries as corroborated by our STM images across the edge (see Fig. 3.3a). The STMimage shows how the graphene grows smoothly over the step without interrup-tion. This assures that a possible strain inside the nanoprism can build up and isnot relieved along grain boundaries. To obtain a more detailed view of the possi-ble strain pattern, we perform additional detailed atomic resolution STM measure-ments. The images taken inside and outside the nanoprisms (Fig. 3.4e) show theexpected (6√3× 6√3)R30◦ modulation with respect to SiC on top of the carbonhoneycomb lattice [59]. However, taking the difference between the two Fouriertransformed images (Fig. 3.4f) reveals a strain pattern inside the nanoprism, witha maximum observed strain of about 3◦. While the strain pattern could not be de-termined for the entire triangle due to limitations during the STM measurements(i.e. large size of the nanoprisms and problems with the stability/cleanliness ofthe tip), Fig. 3.5 shows the differences of Fourier transforms for pristine grapheneand graphene with various lattice deformations and might give some intuition forthe experimentally observed strain pattern. Especially the shear strain in Fig. 3.5dshows some agreement with the experiment with one fixed axis and changes alongthe other two high symmetry directions (compare also Fig. 3.22).In order to confirm if the induced strain pattern indeed leads to flat LLs closeto the Fermi energy, we perform a series of high-resolution ARPES measurements.The experiments were performed at UBC in an ultra-high vacuum chamber equippedwith a SPECS Phoibos 150 analyser with optimal resolutions of ∆E = 6meV and∆k = 0.01A˚ in energy and momentum, respectively, at a base pressure of betterthan p = 7× 10−11 Torr. Photons with an energy of 21.2 eV were provided by aSPECS UVS300 monochromatized gas discharge lamp. Our homebuilt six-axiscryogenic manipulator allows for measurements between 300 K and 3.5 K. Addi-tional datasets were taken at UBC with a second ARPES setup equipped with a53Figure 3.5: Comparison of Fourier transforms for different graphene defor-mations. The figure shows differences between Fourier transforms of pristinegraphene and graphene with various lattice deformations. (a) Isotropic stretch of3%. (b) Uniaxial stretch of 3% along the y-direction. (c) Rotation of 3◦. (d) Shearstrain of 3◦ in the x-direction.54Figure 3.6: Momentum-resolved visualization of LLs. (a) ARPES cut throughthe Dirac cone at the K point at 300 K. The data have been divided by the Fermifunction and symmetrized to compensate for matrix element effects [232]. (b) Cutalong the energy axis integrated around the K point in (a). (c) Second derivative ofthe data in (a) [233]. (d) Inverted second derivative of the data shown in (b) aftersmoothing. (a)–(d) Landau levels (LLs) are indicated by arrows. (e) Summaryof LL data sets, with model fit according to Eqn. 3.1 shown in black; the 95%confidence interval of the fit is shown in grey. Different symbols indicate differentsamples and temperatures: sample A (6 K) [hexagons], sample B (6 K) [squares],sample B 2nd data set (6 K) [stars], sample B (300 K) [diamonds], sample C (6 K)[circles], and sample C 2nd data set (6 K) [triangles]. The position of the Diracpoint (DP) is indicated by the black arrow. Inset: Same data plotted versus√n,giving the expected linear behaviour for LLs in a Dirac material. (f) Sketch ofvarious mechanisms which may lead to ARPES intensity inside the cone. Neitherelectron-phonon coupling nor contamination from bilayer graphene can explain theexperimental findings.55Scienta R4000 analyser and a Scienta VUV5000 UV source with optimal reso-lutions of ∆E = 1.5meV and ∆k = 0.01A˚−1 in energy and momentum, respec-tively, for 21.2eV photons. The samples were annealed at 600◦C for about 2h atp = 1×10−9 Torr and then at 500◦C for about 10h at p = 5×10−10 Torr immedi-ately before the ARPES measurements.ARPES is a momentum- and energy-resolved technique that has proven to be apowerful tool in directly studying the electronic band structures of a vast variety ofquantum phases of matter, from strongly-correlated electron systems and high-Tcsuperconductors [102] to topological insulators and semimetals [221, 222, 234].Yet no study of quantum Hall states has been performed, since ARPES is strictly in-compatible with the application of magnetic fields, as essential crystal momentuminformation carried by the photoemitted electrons would be lost through interactionwith the field. However, this is different for pseudomagnetic fields, as they only in-teract with the Dirac electrons inside the material. We note that while a recently de-veloped momentum-resolved technique compatible with magnetic fields has beenreported [235], it necessarily requires sophisticated heterostructures, physically ac-cessible fields, and is limited to a small sector of the Brillouin zone.Our ARPES data, which – due to the ∼1 mm spot size of the photon source –correspond to the spatial average over unstrained and strained regions of the sam-ple, show the expected Dirac cone as well as new flat bands that gradually mergewith the linear dispersion (Figs. 3.6a and 3.6c). The unequal energy spacing ofthese newly observed bands can be extracted from cuts along the energy directionat the K point (Fig. 3.6b) and their second derivative (Fig. 3.6d). The positions ofthe levels are directly read off the cuts along the energy direction without addi-tional fitting routines. We estimate the accuracy of this procedure to determine thepositions to about 20 meV. By plotting the positions of these bands (Fig. 3.6e), weobserve the distinct√n energy spacing which is a hallmark of LLs for graphene’smassless Dirac charge carriers [2], where n is the integer LL index. The spectrumof LLs in graphene is given by [1]En = sgn(n)√2v2F h¯eB · |n|+EDP (3.1)56Figure 3.7: Fermi velocity and quasiparticle lifetime from ARPES. (a) Thelinear dispersion of graphene (black circles) is fitted linearly (red line) to extractthe Fermi velocity. (b) The extracted binding energy dependent line width (blackcircles) is fitted quadratically (red line) to illustrate the decreasing carrier lifetimeat higher binding energies. The blue dashed line indicates a constant offset due toimpurity scattering.where vF is the velocity of the electrons at the Fermi level, h¯ the reduced Planckconstant, e the electron charge, B the magnitude of the (pseudo-)magnetic field,and EDP the binding energy of the Dirac point. Using the ARPES dispersion map inFig. 3.6a, the Fermi velocity is determined to be vF = (9.50±0.08)×105 m/s. TheFermi velocity can be directly extracted from the ARPES data. The momentum dis-tribution curves at each binding energy are fitted using a Lorentzian with a constantbackground. The dispersion of the band can then be fitted linearly to determine theFermi velocity (see Fig. 3.7a). Fitting our experimental data to Eq. (3.1) as donein Fig. 3.6e, we extract the magnitude of the pseudomagnetic field, which yieldsB = (41± 2)T. Note, the uncertainty results from the fitting procedure and takesinto account the spread of the Landau level positions between different samples andtemperatures (compare Fig. 3.6e). Remarkably, this pseudomagnetic field value ap-pears to be consistent between several samples from cryogenic temperatures (6 K)up to room temperature. The model fit also consistently pinpoints the binding en-ergy of the Dirac point to EDP = (460±10)meV relative to the Fermi level, whichgives us additional confidence in the√n dependence of our data points. The valueagrees well with previous reports on this sample system [57, 236] and is attributed57Figure 3.8: Fit of Landau levels for the exponent. The positions of the Landaulevels and the Dirac point (DP) are fitted to a function of the form f (x) = axb + c(dashed grey), where b is the exponent. For a√n behaviour b = 0.5 is expected.The position of the Dirac point with a binding energy of (0.45±0.02) eV is deter-mined from Fig. 3.12b, where the uncertainty takes into account a possible slightvariation of the position of the Dirac point between different samples. The fit re-sults in a value for the exponent of b = (0.53±0.04).to charge transfer from the SiC substrate to the graphene layer. To gain even furtherconfidence that the extracted Landau level positions and the position of the Diracpoint follow a√n behaviour we fit a function of the form f (x) = axb+c to our data(see Fig. 3.8). The fit yields an exponent of b = (0.53±0.04). Taking into accountthe error bars of the fit, this is compatible with a√n behaviour which would havean exponent of b = 0.5.Additionally, the LLs are only resolved in the upper part of the Dirac cone,closer to the Fermi level. We attribute this effect to the increased scattering phasespace as one moves away from the Fermi level, which reduces the scattering life-time of the carriers. The binding energy dependence of the lifetime of the carrierscan also be directly extracted from the ARPES data. The width of the Lorentzians asa function of binding energy can be fitted quadratically with a constant offset. The58Figure 3.9: Band structure of multilayer graphene. (a) Band dispersions ofgraphene along high symmetry directions for monolayer (purple), bilayer (red),trilayer with ABA stacking (blue), trilayer with ABC stacking (blue dashed), quad-layer with ABAB stacking (green), quadlayer with ABAC stacking (green dashed),and quadlayer with ABCA stacking (green dash-dotted) calculated with a tight-binding model with parameters from [238]. (b) Same band structure as in (a), butas a close up around the K point. (c) Calculated bands overlaid on the experimentalARPES data.linewidth is inversely proportional to the quasiparticle lifetime, thus showing howthe latter decreases as one goes away from the Fermi level (see Fig. 3.7b). Thisis a manifestation of a simple Fermi liquid model. Electrons at the Fermi levelhave a certain lifetime between scattering events dictated by the concentration ofimpurities and defects. As one goes to higher binding energies, the phase space forelectron-electron scattering increases ∝ E2b and the lifetime decreases. We proposethis as the reason why, experimentally, our LLs are only clearly resolved in theupper part of the cone closer to the Fermi level. When the scattering rate at somebinding energy exceeds a critical value above which coherent circular orbits can-not be established, the LL quantization in the ARPES measurement disappears. Wenote that such asymmetric behaviour has been reported before in scanning probemeasurements, and was attributed to a shorter vertical extension of wave functionsat lower energies [86] as well as a reduced quasiparticle lifetime away from theFermi level [237].As for other alternative explanations of the data, we note that while previousARPES studies of graphene on SiC have shown a rich variety of features [66, 239],the signature√n spacing of the new levels (Fig. 3.6e and inset) allows us to distin-59Figure 3.10: Model calculation of strain-induced LLs. (a) Top: Honeycomb lat-tice, with the two sublattices A (red) and B (yellow). The black arrows indicate thesymmetry of the strain pattern. Bottom: Triangular flake with strain-induced pseu-domagnetic field B = 41 T. The colour scale indicates the relative bond stretching.(b) Spectral function for the gapless case with Semenoff mass M = 0meV. (c)Energy cut through the Dirac point (K) of the spectral function in (b). The dashedgrey lines indicate the position of the Landau levels (LL) predicted by Eqn. 3.1.guish the observed effect from other possibilities (Fig. 3.6f). For example, if spec-tral weight inside the Dirac cone arose from the coupling of electrons to phonons[66], it would be limited to characteristic vibrational energies. Similarly, contri-butions from bilayer and higher order graphene layers, which can appear in smallquantities near step edges of the substrate during the growth process [239] (seealso AFM adhesion image Fig. 3.3b), would lead to a manifold of bands, but wouldnot reproduce the observed band structure [238, 240]. The calculated band disper-sions for monolayer graphene, bilayer graphene, trilayer graphene, and quadlayergraphene with their different stacking possibilities are shown in Fig. 3.9. The pa-rameters for the tight-binding model are based on the experimental findings ofOhta et al. [238]. Finally, a recent theoretical study shows that for certain defectsin graphene a smearing out of the Dirac point can occur [241]. This would notexplain the additional flat bands inside the Dirac cone and it should be noted thatsuch defects were not observed during the STM measurements on our samples.60To gain deeper insights on the origin of the observed LLs, we model a region ofgraphene experiencing a uniform strain-induced pseudomagnetic field. We use thesimplest such strain pattern, calculated by Guinea et al. [85], which exhibits thetriangular symmetry of the underlying honeycomb lattice. Using a tight-bindingapproach, we directly simulate a finite-size strained region with open boundaryconditions and armchair edges. In detail, we consider a minimal tight-bindingmodel on the honeycomb lattice with nearest-neighbour hoppings and a sublattice-symmetry breaking Semenoff [13] mass term M:H =−t ∑<r,r ′>(c†A(r)cB(r′)+H.c.)+M(∑rc†A(r)cA(r)−∑r ′c†B(r′)cB(r ′))(3.2)where c†A(r) (c†B(r′)) creates an electron in the pz orbital at lattice site r (r ′) on thesublattice A (B) of the honeycomb lattice, t = 2.7eV, and the nearest-neighbourdistance is a0 = 0.142 nm. We neglect the electron spin, and thus consider effec-tively spinless fermions.We construct a flake in the shape of an equilateral triangle of side lengthL∼ 56 nm. The use of armchair edges ensures that we avoid the zero-energy edgemodes appearing for zigzag edges [1]. We apply the simplest strain pattern respect-ing the triangular symmetry of the problem at hand, namely, the pattern introducedby Guinea et al. [85] which gives rise to a uniform (out-of-plane) pseudomagneticfieldB = 4u0h¯βea0zˆ (3.3)where β ≈ 3.37 in graphene[242], and the corresponding displacement field isgiven byu(r,θ) =(uruθ)=(u0r2 sin(3θ)u0r2 cos(3θ)). (3.4)The hopping parameter renormalization induced by this displacement field is cal-culated using the simple prescription:t→ ti j = t exp[− βa20(εxxx2i j + εyyy2i j +2εxyxi jyi j)](3.5)61where (xi j,yi j)≡ r i− r j is the vector joining the original (unstrained) sites i and j,andεi j =12[∂ jui+∂iu j] (3.6)is the strain tensor corresponding to the (in-plane) displacement field u. Outside thestrained region (which we take as a triangle of slightly smaller length LS ∼ 48 nm),we allow the strain tensor to relax: ε → e− r22σ2 ε , where r is the perpendicular dis-tance to the boundary of the strained region, and σ ∼ 1 nm. We define the lengthscale of the homogeneous magnetic field B to be the diameter of the largest in-scribed circle in the triangle of side LS: λ ≡ LS/√3 ∼ 28 nm. We stress here thatour simulated flakes are much smaller than the experimentally observed triangularfeatures of size ∼ 300 nm. The fact that we nevertheless reproduce the experimen-tal features underlines how the number of observable LLs is limited by the lengthscale of the homogeneous pseudomagnetic field λ , rather than by the size L of thenanoprisms themselves. This length scale could be caused by the more complicatedstrain pattern present in the nanoprisms or be induced by disorder.We then diagonalize the Hamiltonian (Eqn. 3.2) with hopping parameters givenby Eqn. 3.5 to obtain the full set of eigenstates |n〉 with energies En, and computethe momentum-resolved, retarded Green’s function using the Lehman representa-tionGRα(k,ω) =∑n| 〈n|c†α(k) |0〉 |2ω− (En−E0)− iη (3.7)where α = A,B is a sublattice (band) index, and η ∼ 20 meV is a small broadeningparameter comparable to the experimental resolution. We then compute the one-particle spectral function,A(k,ω) =− 1pi∑αIm[GRα(ω,k)](3.8)which is proportional to the intensity measured in ARPES (modulo the Fermi-Diracdistribution and dipole matrix elements). We note that using a finite system intro-duces two main effects in the momentum-resolved spectral function: the appear-62Figure 3.11: Evolution of LLs with increasing uniform pseudomagnetic fields.Calculated spectral function in our triangular flake for fields B = 0, 41, 82 and164 T (from left to right).ance of a small finite-size gap at the Dirac points (in the absence of a magneticfield) and a momentum broadening of the bands (see Fig. 3.11a).We find that the observed LL spectra can be well-reproduced by a triangularflake of side length L = 56 nm (Fig. 3.10a), subject to a uniform pseudomagneticfield B = 41 T over the entire flake (Fig. 3.10a). The maximal strain (or relativebond stretching) reaches around 3%, which is in good agreement with our STMmeasurements. The ARPES data can be simulated by calculating the energy andmomentum-resolved spectral function A(k,ω) of this triangular flake, here shownin Fig. 3.10b and 3.10c. Our simulation clearly reproduces the main features ofthe ARPES data, namely levels that: (i) follow√n spacing in energy; (ii) are flatinside the Dirac cone and merge with the linearly dispersing bands; (iii) becomeless clearly resolved with increasing index n.Features (ii) and (iii) can be understood by comparing the characteristic size ofa Landau orbit ∝√nlB (with the magnetic length lB =√h¯eB ) to the length scale λon which the pseudomagnetic field is uniform. For LLs to exist, an electron on agiven Landau orbit must experience a uniform pseudomagnetic field [242], leadingto the condition√nlB  λ . Hence, for large fields B or large λ , flat bands areexpected across the entire Brillouin zone, whereas Dirac cones are recovered in theopposite limit. In Fig. 3.11, we present the spectral function obtained for M = 0and increasing pseudomagnetic fields B = 0, 41, 82 and 164 T to highlight how63Figure 3.12: Determination of the mass term. (a) ARPES cut through the Diraccone. Orange circles indicate the positions of the fitted Lorentzians. The red lineand the dashed red line indicate linear fits through the orange circles for the upperand lower cone, respectively. The cut is symmetrized around the K point in themomentum direction to remove polarisation effects. (b) The same data as in (a),but fitted to a hyperbola instead. (c) Results for the gap size from the hyperbola fitsfor different ARPES slices along ky. The curve shows the expected half-hyperbolaand the gap size of ∼0.25 eV is given by the minimum. Note, for graphene on SiCa range of mass terms is expected [226]. This means in an ARPES experiment wewould average over areas with zero mass term and hence no gap and areas with afinite mass term and hence a finite gap. The fitting procedure outlined above thusonly yields the largest gap in the observed area.LLs evolve from a Dirac cone when B = 0 to completely flat bands when lB λ .This is analogous to keeping B fixed and increasing the size of the flake, but thelatter method is strongly constrained by numerical resources. Here lB = 4.0, 2.8and 2.0nm at B = 41, 82 and 164 T respectively, whereas λ ∼ 30 nm.The bands observed in the ARPES data can thus be understood as LLs, where theorbit size is only somewhat smaller than λ : by comparing the experimental dataand the model calculation, we estimate lB ∼ 4 nm and λ ∼ 30 nm. Furthermore,since the size of Landau orbits grows as∼√|n|, eventually it becomes comparableto λ , explaining why levels with higher index n are less clearly resolved.However, our simple model (Figs. 3.10b and 3.10c) consistently exhibits asharp zeroth LL (LL0), which is absent from the ARPES data. This discrepancy issurprising, since LL0 is known to be stable against inhomogeneities of the magnetic64field as well as against disorder, as long as the latter preserves the chiral symmetryof graphene [243]. Below, we provide a possible mechanism that broadens LL0without significantly affecting the higher LLs. It has been argued that graphenegrown on SiC is subject to a sublattice-symmetry-breaking potential arising fromthe interaction with the substrate [226]. The minimal theoretical model describ-ing this effect, which acts as a staggered potential between sublattices A and B,is the so-called Semenoff mass M [13]. Here we briefly discuss the effect of aSemenoff mass [13] M on pseudo-LLs and show that a uniform Semenoff masscannot explain the observed spectrum. Starting from the linearly dispersing bandsin the Dirac cone without any magnetic fields, a mass term opens a gap at the Diracpoint. The size of the gap is equal to twice the size of the mass term M. Experi-mentally, it manifests in our ARPES cuts through the Dirac point by extending thelinear dispersions of the lower and upper cones, for both sides with respect to theK point (Fig. 3.12a), in that these extrapolations do not meet in a single point, butare offset from each other. To accurately determine the size of the gap, we fit twoLorentzians with a constant background to momentum distribution curves in theupper and lower cones. The energy range of the fit is selected to avoid the promi-nent LLs. A hyperbola is then fitted to the bands (Fig. 3.12b) to determine top andbottom of the two bands, and in turn the gap size. The procedure is repeated forseveral cuts through the Dirac cone along the ky direction. The results are sum-marised in Fig. 3.12c and the mass term is equal to half of the minimal gap size(∼0.25 eV). This is comparable to the ∼0.26 eV gap observed in the same samplesystem by Zhou et al. [226].Next, we describe the effects of a mass term on a Dirac dispersion includingmagnetic fields. In short, the mass term opens a gap at the Dirac point and shiftsthe LL spectrum for n 6= 0 to [244]:En = sgn(n)√2ev2F h¯B · |n|+M2+EDP. (3.9)But note, that Eqn. 3.9 is not properly defined for n = 0 – to understand whetherLL0 is shifted to +M or −M (in valleys K and K′), we have to distinguish betweenreal magnetic fields, which break time-reversal symmetry, and pseudomagneticfields, which preserve time-reversal symmetry. For real magnetic fields [244], LL065Figure 3.13: Sketch of pseudo-LLs with Semenoff mass. Depending on the signof the mass term M, the zeroth LL (LL0) gets shifted to the upper or lower part ofthe cone. The spectrum is identical for valleys K and K′, because pseudomagneticfields preserve time-reversal symmetry. Higher LLs only get pushed away slightlyfrom the Dirac point.has opposite energy ±M at K and K′. For pseudomagnetic fields, in order to pre-serve time-reversal symmetry, the spectrum must be identical in both valleys, andthe energy of LL0 is determined by the sign of M, so for n = 0 we simply getELL0 = EDP±M. This is illustrated in Fig. 3.13 for different signs of the massterm, where LL0 either shifts to the top of the lower cone (M < 0) or the bottom ofthe upper cone (M > 0).Our numerical simulations clearly show this behaviour (Fig. 3.14), but thereis one additional caveat. The total pseudomagnetic flux must be vanishing in ourflake by construction, as we require the strain to relax at the edges of the flake.This requirement generates a region near the boundaries of the strained area witha pseudomagnetic field of the reversed sign. This region hosts a LL0 at an en-ergy inverted with respect to the LL0 coming from inside the strained area. This isvisible in our calculations as weaker and more broadened (in momentum) levels,indicated by red arrows in Figs. 3.14b and 3.14c. Note that experimentally, a sim-ilar scenario is natural on our graphene on SiC samples as well. The strain insidethe nanoprisms needs to relax away from the feature, thus creating an area with aninverted pseudomagnetic field.To check if a uniform mass term of about the determined size can explain ourfindings, we fit the observed LLs to Eqn. 3.9 (see Fig. 3.15). While this model pro-66Figure 3.14: Calculation of pseudo-LLs with Semenoff mass. Calculated spec-tral function in our triangular flake with a uniform pseudomagnetic field B = 41 Tand Semenoff masses M = 0, M = −135 meV, M = +135 meV, and averaged inthe interval M ∈ [−135,135]meV (from left to right). The positions of the LLs forthe different cases are indicated, as well as the much weaker LL0 from the areasurrounding the strained flake (red arrows in (b) and (c)).Figure 3.15: Model fit with constant mass term. Fit of the observed LLs toEqn. 3.9. Note the shifted indices for the LLs in this scenario. It places the Diracpoint at a binding energy of 390 meV with M = 150meV, compared to 450 meVobtained from the fit to Eqn. 3.1 without a mass term.67Figure 3.16: Calculation for a uniform mass distribution. (a) Spectral functionaveraged over a uniform distribution of Semenoff masses M ∈ [−135,135]meV.(b) Energy cut through the Dirac point (K) of the spectral function in (a). Theshaded grey area indicates the broadening of the LLs predicted by Eqns. 3.1 and3.9.duces a qualitatively good fit with M = 150meV, it places the Dirac point at abinding energy of 390meV, which is inconsistent with the experimental observa-tions (compared to 450meV obtained from the fit to Eqn. 3.1 without a mass term).This means, a uniform mass term M cannot explain the ARPES data.Therefore, we postulate that the mass term M varies on a length scale muchgreater than the magnetic length lB ∼ 4 nm, but smaller than the ARPES spot size(∼1 mm). The variation can take place either from nanoprism to nanoprism, orwithin a given nanoprism, if it is tied to the length scale of the uniform pseu-domagnetic field λ . In the former scenario, we can approximate the effect ofthe slowly-varying mass term M by averaging over the spectral function obtainedwith different fixed M (such as those shown in Figs. 3.14b and 3.14c). This isshown in Figs. 3.16d and 3.16e for a uniform distribution in the interval M ∈[−135,135]meV. As evident from Eqn. 3.9, the distribution of mass terms affectsLL0 most, while merely contributing an additional broadening to the higher levels.Note that, as observed experimentally, the variation of the mass term is not limitedto the strained areas, but instead is a property of the whole sample; as a result,ARPES always picks up a spatial average of strained areas with LLs and unstrained68Figure 3.17: Measurement of the Raman spot size. (a) Image of the laser spotwith the ×50 objective. (b) Line cut through the laser spot. A fit to a Gaussianprofile (red) yields a FWHM of about 6 µm. (c) Image of the laser spot with the×100 objective. (d) Line cut through the laser spot. A fit to a Gaussian profile(red) yields a FWHM of about 3 µm. The flat parts of the line cuts in (b) and (d)indicate a saturation of the detector at the center of the laser spot. These pointswere omitted for the purpose of the fits.areas with the usual Dirac cone dispersion, both having the same distribution ofmass terms and corresponding Dirac point gaps. This phenomenological modelcompletely smears out LL0, while only slightly broadening the other levels (seeFig. 3.14d) and is in good agreement with the experimental data and may renewinterest in the variation of the mass term in this sample system [226].To support our idea of a mass term that varies on a length scale larger than thenanoprisms, we perform Raman spectroscopy measurements on our samples. The69measurements were taken at room temperature and in air on a HORIBA LabRAMsystem with a helium-neon laser (632.8 nm wavelength).In previous experiments, it could be shown that a shift accompanied by achange in the width of the so-called graphene 2D feature indicates a change in thenumber of graphene layers, while a shift without a change in the width of grapheneRaman features can indicate a modification of the interaction between grapheneand the underlying substrate [213]. The lateral resolution of a Raman measure-ment is determined by the spot size of the laser beam on the sample. Images ofthe laser spots and an analysis of the profiles for our system for the ×50 and ×100objectives are shown in Fig. 3.17. Lateral resolutions of 6 µm and 3 µm respec-tively could be determined from Gaussian fits. Using the ×100 objective, Ramanmaps of the graphene 2D peak were acquired (see Fig. 3.18). The 2D feature canbe fitted with a Gaussian to extract its amplitude, position, and width. Step edgesin the substrate with small contributions from bilayer graphene should show up asline features with a shift in the peak position as well as a change in the peak width.No features consistent with this are observed, indicating that the lateral resolutionof the experiment is probably not good enough to resolve the step edges.Additional measurements were taken as line measurements along the x-direction(see Fig. 3.19) as well as the y-direction (see Fig. 3.20), which allow us to coverlarger lateral distances compared to the map measurements. Also here a Gaussianwas used to fit the 2D peak in each spectrum to determine the peak position as wellas the peak width. For the y-direction, no distinct features are visible. In contrastto that is the measurement along the x-direction. Here the peak position shows aperiodic behavior, which is mostly independent of the peak width (see Fig. 3.19band Fig. 3.19c).The peak positions of the 2D feature along the x-direction can be fitted with asine function, as done in Fig. 3.21b. This hints at the possibility that the interactionbetween the graphene layer and the SiC substrate is modulated with a period ofabout 20 µm. If one makes the reasonable assumption that the mass term oscillatesin accordance with the graphene-substrate interaction, this supports the idea of avarying mass term between individual nanoprisms. We note that our STM mea-surements are carried out on much smaller length scales compared to the observedmodulation with Raman. Hence, it is not surprising to not see any evidence in this70Figure 3.18: Raman maps of the graphene 2D peak. Raman spectra were takenon a 20 µm × 20 µm grid (20 px × 20 px) with the ×100 objective. The graphene2D peak was fitted to a Gaussian, which allows the extraction of the peak amplitude(a), peak position (b), and peak width (c).71Figure 3.19: Raman line spectrum x-direction. (a) Raman line spectrum of theso-called graphene 2D feature along 60 µm in the x-direction. Each spectrum wasfitted with a Gaussian to extract the peak position (b) and peak width (c).regard in those measurements. Currently we can only speculate about the under-lying origin of the modulation. Typical length scales for superstructures betweengraphene and SiC are much shorter, but a relation to the step edges on the substrate(compare Fig. 3.1a), which have a spacing of several µm, seems possible.Finally, we compare the strain model used in the calculations to the experimen-tally observed strain geometry, as far as possible (see Fig. 3.22). For that purposewe select an area in each of the three corners of the nanoprism and compare their72Figure 3.20: Raman line spectrum y-direction. (a) Raman line spectrum of thegraphene 2D feature along 60 µm in the y-direction. Each spectrum was fitted witha Gaussian to extract the peak position (b) and peak width (c).atomic lattice to an unstrained area outside the nanoprism. This is done using ad-ditional atomically resolved STM images and their Fourier transforms. Comparingthe results to the theoretical results, we see that both agree well in terms of overallmagnitude of the strain and the 2pi3 symmetry. While in the experiment the ampli-tude of the strain rotates as one goes from corner to corner, it is the phase of thestrain with a constant amplitude in the calculations. This finding might motivatefuture theoretical studies in the area of pseudomagnetic fields.73Figure 3.21: Raman line spectra with fit. (a) Raman line spectra along the xdirection around the graphene 2D peak. (b) The peak positions of the 2D peak(black points) are determined with a Gaussian fit of the individual spectra in (a).The peak positions can be fit with a sine function with a period of about 20 µm(red).In summary, this study provides the first demonstration of the room tempera-ture strain-induced quantum Hall effect in graphene on a wafer-scale platform, aswell as the first direct momentum-space visualization of graphene electrons in thestrain-induced quantum Hall phase by ARPES, whereby the linear Dirac dispersioncollapses into a ladder of quantized LLs. This opens a path for future momentum-resolved studies of strain-induced, room temperature-stable topological phases ina range of materials including Dirac and Weyl semimetals [245–247], monolayertransition metal dichalcogenides [248], and even nodal superconductors [249, 250],all under large, potentially controllable pseudomagnetic fields. Importantly, thesesystems will feature time reversal invariant ground states – otherwise impossiblewith a true magnetic field – and may act as future building blocks for pseudospin-or valleytronic-based technologies [251]. In light of the recently discovered un-conventional superconductivity in “magic angle” twisted bilayer graphene [74, 83],strain-induced pseudomagnetic fields likewise raise the possibility of engineeringexotic time reversal symmetric variants of correlated states including superconduc-tivity in LLs [252] and fractional topological phases [253]. Our results lay the foun-74Figure 3.22: Comparison of experimental and model strain. (a) – (d) STMtopography images with Vsam. = 30mV and Itun. = 2 pA taken at each corner ofthe nanoprism and outside. The Fourier transform of each image is shown as aninset. (e) Schematic showing the position of the measurements with respect to thenanoprism. (f) – (h) Difference of the Fourier transforms between the strained areasinside the triangle (top, bottom and side) and the unstrained area outside. (i) – (k)Difference of the Fourier transforms between the strained areas inside the triangle(top, bottom and side) and unstrained graphene for the model calculation.dations for bottom-up strain-engineering of novel macroscopic quantum phases atroom temperature and at the technologically relevant wafer-scale.75Chapter 4Correlated electron physics ingadolinium intercalated grapheneQuantum materials include several classes of materials in which the interaction be-tween electrons leads to a complex phase diagram. They feature phenomena likeunconventional superconductivity, pseudogaps, or density wave orders. Unravel-ling the origin and interplay of these quantum phases represents one of the greatchallenges in physics today and is key in predicting the future design of quantummaterials and their translation from fundamental research to applications. In lightof the recent discovery of a Mott insulating phase and unconventional superconduc-tivity in so called “magic angle” graphene, we investigate the novel, tailor-made,and wafer-scale sized quantum material based on ultra-highly doped graphene cou-pled to an ordered monolayer of gadolinium. We demonstrate correlation-inducedflat bands, a temperature-dependent pseudogap, and signatures of a density waveorder up to room temperature.The physics of correlated electrons remains at the forefront of formidable chal-lenges in science and describes materials in which the interaction between elec-trons cannot be treated as an averaged background. Materials in the field typicallyfeature a complex phase diagram as a function of electronic doping, mechanicalpressure, magnetic field, and temperature, displaying remarkable phenomena likeunconventional superconductivity, a pseudogap phase, antiferromagnetism, or den-sity wave orders (see Fig. 4.1a). Prominent examples include the cuprates [84]76Figure 4.1: Introduction to Gd-intercalated graphene. (a) Schematic exem-plary phase diagrams of prominent correlated electron physics material classes asa function of doping (cuprates and iron-based) or pressure (organic compoundsand heavy fermions). All show a complex interplay between different quantumphases such as superconductivity (SC), charge order (CO), pseudogaps (PG), spin-Peierls (SP) phases, or antiferromagnetic order (AF). Adapted from [254]. (b)Electronic band structure of graphene. The linearly dispersing Dirac cones touchat the corners of the Brillouin zone (BZ), while the saddle point halfway betweenneighboring corners along the BZ edge lead to a Van Hove singularity (VHS). Highsymmetry points of the hexagonal BZ are indicated. (c) Illustration of the structureof Gd-intercalated graphene. The Gd atoms (orange) sit between the topmost Siatoms of the substrate (red) and the graphene monolayer (grey). The size of theunit cell increases from (13×13) units of graphene (blue) to (13√3×13√3)R30◦units of graphene (yellow) for the whole sample system of substrate, Gd atoms, andgraphene. (c) is adapted from the dissertation of S. Link, one of our collaboratorson this project at MPI Stuttgart [255].and iron-based [256] superconductors, organic superconductors [257], and heavyfermion compounds [258]. Understanding the manifold of quantum phases andtheir interplay is often complicated by equally complex chemical and structuralproperties. Hence, a search for new platforms for correlated electron physics ismandated. Graphene, a two-dimensional honeycomb layer of carbon atoms, is oneof the most widely studied materials over the past 15 years [1, 2] and has recentlyemerged as a prime candidate after the reported discovery of a Mott insulatingphase [83] and unconventional superconductivity [74] in bilayer graphene. It wasproposed that when the two graphene layers are twisted with respect to each otherby the so called “magic angle”, flat bands close to the Fermi level lead to the en-77hancement of electronic correlations. An alternative approach to realize flat bandsin graphene is large electronic doping up to the M point, where the Dirac disper-sions of two neighbouring K points meet (see Fig. 4.1b).Exploiting this route, we investigate a new material based on intercalated epi-taxial graphene on SiC [57]. The samples were grown on commercial 6H-SiCwafers. The substrates were hydrogen-etched prior to the growth under argon at-mosphere. Details are described by S. Forti and U. Starke [227]. Gadolinium is de-posited from an electron-beam evaporator while the samples are held at 600◦C. Thesamples are subsequently flashed to 1000◦C to complete the intercalation process.Samples are characterized by low energy electron beam diffraction (LEED), lowenergy electron microscopy (LEEM), and X-ray photoelectron spectroscopy (XPS)(for details see dissertation of S. Link, one of our collaborators on this project atMPI Stuttgart [255]). For ex-situ experiments, the samples are capped with a thinbismuth layer to prevent oxidation. The cap can be removed by annealing the sam-ples in ultra-high vacuum (UHV). Using gadolinium as an ordered intercalant, weare able to induce the required doping levels. A single layer of Gd atoms arrangedin a triangular lattice sits between the topmost Si atoms of the SiC(0001) substratebelow and the monolayer graphene on top. The structure leads to an enlarged unitcell, covering (13√3×13√3)R30◦ graphene unit cells, as determined from LEEDmeasurements [255] and illustrated in Fig. 4.1c. LEED images allow the determi-nation of the sample quality, orientation, and atomic structure.Angle-resolved photoemission spectroscopy (ARPES) has proven to be a pow-erful tool for the study of novel quantum materials and their intertwined phasediagrams [102, 103]. We use ARPES to directly visualize the electronic bandstructure of our material close to the Fermi level. Experiments with He I pho-tons (21.2 eV) were performed at UBC in a ultra-high vacuum chamber equippedwith a SPECS Phoibos 150 analyzer with optimal resolutions of ∆E = 6 meV and∆k = 0.01 A˚−1 in energy and momentum, respectively, at a base pressure of betterthan p = 7×10−11 Torr. Photons were provided by a SPECS UVS300 monochro-matized gas discharge lamp. Our homebuilt six-axis cryogenic manipulator al-lows for measurements between 300 K and 3.5 K. The samples were annealed at600◦C for about 2 h at p = 1× 10−9 Torr and then at 500◦C for about 10 h atp = 5× 10−10 Torr immediately before the ARPES measurements. Experimental78Figure 4.2: Electronic structure measured with angle-resolved photoemissionspectroscopy (ARPES). (a) Symmetrized Fermi surface (FS) measured with He Iphotons (21.2 eV). (b) ARPES cut through the Dirac point as indicated by the yel-low line. The Dirac point is shifted to a binding energy of about−1.6 eV due to theelectron doping of the intercalated Gd atoms. (c) ARPES cut along the KMK di-rection as indicated by the yellow line. Around the M point the experiment revealsare remarkably flat dispersion, not expected by a nearest-neighbour tight-bindingcalculation (green), where the band position was shifted down in energy to reflecthighly doped pristine monolayer graphene. (d) Schematic illustration of the FSevolution for graphene as a function increased electron doping. Starting with twocircular electron pockets centered at the corners of the Brillouin zone (BZ) (top),increased doping leads to trigonal warping (middle). Further doping induces a Lif-shitz transition of the FS topology to a single hole pocket centered around the centerof the BZ (bottom). Panels (b) – (d) were adapted from the dissertation of S. Link,one of our collaborators for this project in the group of U. Starke at MPI Stuttgart[255].data shown in Figs. 4.2b+c were acquired at the I4 beamline at the MAX III syn-chrotron facility in Lund, Sweden (for details on these measurements see [255]).Looking at the Fermi surface (FS) for graphene with increasing electron doping,one expects two circular electron pockets centered on the K points of the Brillouinzone (BZ) which show triangular warping as the electron density increases. Finally,at the Lifshitz transition, the two pockets merge and the FS topology changes to asingle hole pocket centered on the Γ point of the BZ. This is illustrated in Fig. 4.2d.The ARPES data show, that it is possible to reach the required doping levels inmonolayer graphene through the intercalation of gadolinium (Figs. 4.2a–c). TheFS shows the expected triangular warping and merging of the two electron pockets.79Note that for He I photons (21.2 eV) (Fig. 4.2a) the spectral weight at the M pointis suppressed because of matrix element effects. The temperature dependent datataken at the M point in Fig. 4.12 shows that there is indeed spectral weight at thispoint and the two electron pockets are merging and are starting to form a singlehole pocket. In addition, data taken by our collaborators at MPI Stuttgart [255] ata synchrotron facility with variable photon energy (Fig. 4.2c) show more clearlythat the dispersions from neighbouring K points are touching. The Dirac point isshifted down to a binding energy of about 1.7 eV (Fig. 4.2b). Around the M point,the measurements reveal a flat band at the Fermi level, strongly renormalized fromthe calculated dispersion for highly doped pristine graphene (Fig. 4.2c). Such abehaviour has been observed before in graphene samples doped with calcium andpotassium and was attributed to the proximity of the Van Hove singularity to theFermi level in combination with many-body interactions [65]. More recently thiswas also reported for graphene samples on iridium substrates doped with caesium[259]. Here the importance of band folding and hybridization in the system wasstressed.Compared to the tight-binding calculation, quadratic fits show an enhancementof the effective mass of the charge carriers by about a factor of 30 (m∗KMK ≈ 24m0)along the KMK direction, while perpendicular to that along the ΓMΓ direction,the effective mass remains almost unchanged (m∗ΓMΓ ≈ 0.2m0) (see Fig. 4.3). Todetermine the effective mass of charge carriers around the M point, the band disper-sion as measured by ARPES can be used (compare Fig. 4.2c). Energy distributioncurves (EDCs) are fitted to extract the position of the band along both the KMKand the ΓMΓ directions. To allow the comparison between the two high symme-try directions, the “replica” band, which resides about 0.25 eV below the Fermilevel, was used. The bands are then fitted to a quadratic function, which givesaccess to the curvature of the bands and thus the effective carrier mass via the rela-tion me f f = h¯2/| ∂ 2E∂k2 | (see Fig. 4.3). This yields an effective mass for the hole-likecarriers along KMK of (24.2±1.1)m0 and an effective mass for the electron-likecarriers along ΓMΓ of (0.20±0.02)m0. Note that we are comparing the absolutevalues of the effective masses. Because of the curvature of the bands for electronsand holes one expects a positive value for the effective mass of electrons and anegative value for the effective mass of holes. The errors are determined from the80Figure 4.3: Analysis of the effective mass of the charge carriers. (a) Banddispersion along the KMK direction around the M point (indicated by the greenline in the schematic Brillouin zone) as determined from fits to energy distributioncurves (EDCs). The data points are fitted to a quadratic function (red) to extract theeffective band mass. (b) Band dispersion along the ΓMΓ direction around the Mpoint (indicated by the green line in the schematic Brillouin zone) as determinedfrom fits to EDCs. The data points are fitted to a quadratic function (red) to ex-tract the effective band mass. The “replica” band about 0.25 eV below the Fermilevel (compare Fig. 4.2c above) was used for the analysis to allow the comparisonbetween both high symmetry directions.uncertainties of the quadratic fit to the band dispersion. These values can be com-pared to the tight-binding model for pristine graphene. The next-nearest neighbourmodel predicts carrier masses of 0.24 m0 along ΓMΓ and 0.73 m0 along KMK.Hence, a mass enhancement of roughly a factor of 30 is experimentally observedalong KMK, while along ΓMΓ the effective mass remains mostly unchanged com-pared to the theoretical prediction. This creates a highly anisotropic picture withheavy hole-like carriers along KMK and comparatively light electron-like carriersalong ΓMΓ.The area of the Fermi surface (AFS) can be set in relation to the area of the Bril-louin zone (ABZ) to obtain information about the number of charge carriers per unitcell (Nuc) using the equation Nuc = 2 · AFSABZ . The factor of two arises from the spindegeneracy of the bands in graphene. We are using the area of the graphene Bril-louin zone as reference, so that the resulting number of electrons is calculated pergraphene unit cell. Luttinger’s theorem [260] thereby guarantees that the area of81the Fermi surface is conserved even under the influence of the many-body interac-tions present in the sample systems. We extract the carrier density in the system to(5.1±2.0)×1014 cm−2. This number is also supported by transport measurementsby our collaborators in the groups of U. Starke and J. Smet at MPI Stuttgart whomeasured the Hall resistance of the samples as a function of magnetic field. Thiscorresponds to a doping of 0.8± 0.2 electrons per Gd atom, agreeing well withthe expectation that two electrons per Gd atom are needed to saturate the danglingSi bonds of the substrate to establish a quasi-freestanding monolayer of graphene[60].Hence, gadolinium is in the Gd3+ configuration, leaving the half-filled f -orbitalsas the only open shell. In combination with the well-known Coulomb repulsion ofthe f -electrons in gadolinium leading to a split of the f -bands [261, 262], this natu-rally explains the lack of Gd states in the ARPES data around the Fermi energy. Thedescribed picture is supported by synchrotron photoemission measurements by ourcollaborators in the group of U. Starke at MPI Stuttgart [255]. Their experimentallows the identification of the Gd lower Hubbard f -states through a highly pho-ton energy-dependent photoemission cross-section [263]. The data show spectralweight around the expected 9 eV binding energy, which exhibits a strong resonanceenhancement across a Gd absorption edge (Gd N5 edge around 150 eV) [255].After describing the electronic structure of the material, we now turn to the pos-sible ordering phenomena of the intercalated gadolinium atoms. Resonant energy-integrated X-ray scattering (REXS) has been an invaluable tool in elucidating emerg-ing ordering phenomena in a range of materials with high sensitivity and chemicalselectivity [144, 161]. Our experiments were performed at the resonant soft X-rayscattering (RSXS) endstation of the REIXS beamline of the Canadian Light Sourcein Saskatoon. The photon energy can be changed from 100 eV to 2500 eV with anenergy resolution E/∆E > 5000. The endstation is equipped with a 10-motionultra-high vacuum diffractometer and a closed-cycle cryostat for variable sampletemperatures from 22 K to 380 K. The pressure during experiments is better thanp = 5× 10−10 Torr. Samples were annealed in situ to about 500◦C for 8 h imme-diately before the experiments. Samples were aligned using a SiC substrate Braggdiffraction spot in situ and low energy electron diffraction (LEED) measurementsduring sample growth. Despite our material only containing a single atomic layer8283Figure 4.4: Ordering phenomena revealed by resonant energy integrated X-ray scattering (REXS). (a) Photon energy-dependent X-ray absorption at theGadolinium M4/5 edge. Both transitions (3d5/2 and 3d3/2) are clearly visible, de-spite the material incorporating only a single layer of Gd atoms. (b) Momentum-resolved measurements along the graphene direction on resonance (red) and offresonance (blue and green) reveal a Gd ordering vector around 0.07 A˚−1 (red star).The inset shows the same data with the background removed. The photon energiesfor the three scans are indicated by red, green, and blue circles respectively in panel(a). (c) Temperature-dependent measurements of the Gd ordering vector show astrong stability against thermal fluctuations. (d) REXS intensity as a function ofin-plane momentum resonant to the Gd M5 edge. The data has been interpolatedand symmetrized based on 41 slices taken in a range of about 120◦. (e) Resonancemap across the Gd M5 edge for two prominent features along the negative kx direc-tion. A strong resonance enhancement for both peaks is evident. (f) Comparisonof the experimental resonance behavior of the peak indicated by the white box inpanel (e) after background subtraction (black) to the expected behavior for a smallperiodic modulation of the Gd lattice positions (red).Figure 4.5: Fitting of peaks in the parallel momentum plane. Ordering vectorsextracted from the in-plane momentum REXS map in Fig. 4.4d. Each measuredslice from the REXS experiment is fitted to a background and Gaussian peaks. Fromthere, the peak positions, intensities, and widths can be read out. Peak intensitiesare represented by the size of the circles and different colours correspond to thepeak widths.84of Gd, we can clearly identify the resonant M4/5 edge around 1200 eV photon en-ergy (Fig. 4.4a). Along the graphene direction, e.g. the direction along whichthe structural graphene Bragg peak would be expected at higher momentum, weresolve an ordering peak resonant to the Gd edge (Fig. 4.4b). This feature is sur-prisingly stable up to 380 K (see Fig. 4.4c). The coherence length of the phase canbe estimated by computing the inverse half width at half maximum of the peak[151] and comes out to (72.5±2.6) A˚. To obtain a full picture of momentum space– as far as it is accessible with the given photon energy at the gadolinium reso-nance – we rotate the sample in small increments to measure slices along differentdirections in the parallel momentum plane. The results are summarized in 4.4d.We are able to reveal a number of ordering phases, which mostly follow the ex-pected 60◦ or 120◦ symmetry. After a background subtraction, the features can befitted with Gaussians to extract their position, width, and intensity. The results aresummarized in Fig. 4.5. We note that the photon energy at the Gd M4/5 edge limitsus to about 1 A˚−1 in momentum space. The experiment is therefore more suitablefor longer wavelength orders, which in turn appear at small momenta in scatteringexperiments. For comparison the (√3×√3)R30◦ superstructure observed withLEED [255] corresponds to a scattering vector length of 1.7 A˚−1 and is thus notdetectable in our X-ray scattering experiments.To gain insight into the origin of the observed ordering vectors, we measure de-tailed resonance maps across the gadolinium M5 edge (see Fig. 4.4e). The extractedphoton energy-dependent resonant scattering behaviour can then be compared tocalculated scattering factors based on parameters from the X-ray absorption spec-troscopy data. For a theoretical description of the resonance behavior of the ob-served REXS features, we use the software package Quanty [264–266]. It allowsmodelling of the 3d to 4 f transition at the Gd M4/5 edge including all relevantspin-orbit and multipole Coulomb interactions. In a first step, the model parame-ters are optimized to match the results of the X-ray absorption spectroscopy (XAS)data (see Fig. 4.6a). Using these parameters, the full complex isotropic ( f 0) andmagnetic ( f 1) scattering factors can be calculated (see Fig. 4.6b). Note that theimaginary part of f 0 is the XAS signal. In a final step, the scattering factors areused to compute the energy-dependent resonant scattering for different orderingphenomena (see Fig. 4.7). We compare magnetic ordering (I(Ein) ∝ | f 1(Ein)|2),85Figure 4.6: Theoretical description of the X-ray absorption spectroscopy(XAS) signal at the Gd M4/5 edge. (a) The parameters of the model (red) areoptimized to match the experimental data (black). (b) Using the parameters from(a), the real (black) and imaginary (red) parts of the isotropic ( f 0) and magnetic( f 1) scattering factors can be calculated.charge ordering (I(Ein) ∝ | f 0(Ein−∆E)− f 0(Ein +∆E)|2), and a small periodicmodulation of the lattice (I(Ein)∝ | f 0(Ein)|2) [267]. The experimental data clearlyagrees best with the lattice modulation (see Fig. 4.4f and Fig. 4.7). We note that theemergence of density wave orders is usually accompanied by such a modulation[268–270] and that the charge response might be covered by the lattice responseof f -electrons close to the atomic core. Hence, we observe signatures of densitywave type orders in the material.Further evidence for the possibility of density wave order comes from the po-larization dependence of the resonant signal (see Fig. 4.8). The signal is strong86Figure 4.7: Energy-dependent resonant scattering at the gadolinium M5 edgefor different ordering phenomena based on the complex scattering factorsfrom Fig. 4.6. (a) Comparison of experimental data (black) to theory based onmagnetic ordering (red). (b) Comparison of experimental data (black) to theorybased on charge ordering (red). (c) Comparison of experimental data (black) totheory based on a small periodic lattice modulation (red).Figure 4.8: Polarization dependence of REXS signal. The observed peaks reso-nant to the Gd edge show a strong dependence on the incident photon polarization.The light comes in at a grazing angle of 15◦, so that the linear vertically polarizedphotons (red) are mostly oriented in the sample plane, while the linear horizontallypolarized photons (blue) are mostly oriented out of the sample plane.87for incident photons with a polarization alignment in the sample plane, while it ismostly suppressed for photons with a polarization aligned out of the sample plane.This is similar to what has been observed for density wave phases in the cupratesuperconductors [271]. We note that up to now we were only able to analyzethe resonance behaviour and hence the scattering origin of the single peak shownabove. One could argue that the similar temperature dependence of peaks hintsat the same scattering origin, but a more extensive analysis in future experimentsseems mandated.Next, we briefly discuss the magnetic properties of Gd-intercalated graphene.Applying Hund’s rules to the Gd3+ configuration leads to a 8S7/2 ground state withzero orbital momentum and a 7/2-spin magnetic moment. Our collaborators at MPIStuttgart have measured the magnetization of the gadolinium intercalated graphenesamples to look into the possibility of an ordering of the magnetic moments [255].The material shows no magnetization at zero magnetic field, excluding a simpleferromagnetic picture. Note that bulk gadolinium enters a ferromagnetic phasearound room temperature [272]. For large magnetic fields the magnetization peratom approaches 7µB, consistent with the Gd3+ picture. Further, the magnetizationis positive for positive magnetic fields and negative for negative magnetic fields,which indicates that the material is not dominated by diamagnetic interactions. Foran ideal paramagnet, the magnetization is expected to follow the Brillouin functioninstead. The experimental magnetization – measured at a temperature of 2 K –does not follow the Brillouin function at 2 K. This could indicate that an interactionbetween the Gd magnetic moments is present, hindering the atomic moments fromaligning in the external magnetic field. But it should be noted that an orderedphase might have a short coherence length with fluctuating character. We also notethat an identical magnetization curve was measured for two different angles ofincidence for the X-ray magnetic circular dichroism (XMCD) measurements [255].Finally, we have to keep in mind that the gadolinium atoms sit on a triangularlattice (compare Fig. 4.1c), and thus a highly degenerate magnetic ground state dueto frustration of the spin moments has to be considered.One possibility to resolve the frustration of a triangular lattice is a Heisenberg-type spin ordering. Instead of two distinct spin orientations (up and down) in asimple antiferromagnetic order, this model incorporates three different spin orien-88Figure 4.9: Possible model of antiferromagnetic order in Gd-intercalatedgraphene. (a) Heisenberg-type ordering with three different spin orientations(blue, red, and yellow) of the Gd unit cell including graphene and the SiC sub-strate. The original unit cell is indicated in dashed yellow, the unit cell with theadditional magnetic order in dashed purple. (b) Possibility for the alignment ofeach Gd magnetic moment within a unit cell. The in-plane orientation of the spinsis colour-coded and shown here for a spin-up cell (red) as shown in (a). (c) Thevortex-like structure leads to a finite out-of-plane spin component. The simplestpossibility for this component is shown here with positive out-of-plane spin in redand negative out-of-plane spin in blue.89tations (each 120◦ offset from the next one). Applying the Heisenberg model to theunit cells of Gd-intercalated graphene is illustrated in Fig. 4.9a. Here the large unitcells are ordered in the 120◦ pattern instead of individual atoms. One possibility forthe alignment of the Gd spins within a cell is depicted in Fig. 4.9)b for the in-planecomponent and in Fig. 4.9)c for the out-of-plane component. The pattern resemblesthe vortex-like structure of skyrmions, which have been predicted and observed inother condensed matter systems [273–278]. We stress that the shown model isonly one possibility of a magnetic order on a triangular lattice and the detailed spinconfiguration of the Gd moments is presently not known. Finally, looking at themomentum distribution of some peaks in Fig. 4.4d and Fig. 4.5, a large radial ex-tension in the parallel momentum plane is observed. A similar behaviour with aglass-like state has recently been discussed in terms of an antiferromagnetic phasetransition in a cuprate superconductor sample system [279].The next question is, which quantum phases can be induced as a result of apossible coupling between the flat-band electrons in graphene and the orderingphenomena of the intercalated Gd atoms? A natural starting point is the cross-ings between the original FS and the reconstructed FS due to the Gd order. Thisis illustrated for the tight-binding-based band dispersion for highly-doped pristinegraphene in Fig. 4.10a. Note, for simplicity we are limiting ourselves here to thecase of next nearest neighbour hopping terms in graphene. For a complete descrip-tion of the band structure additional terms can be considered, e.g. next nearestneighbour hopping terms in graphene or terms describing the interaction with thegadolinium atoms and the substrate. For this purpose we only consider the orderingvector with longest coherence length (see Figs. 4.4b+c), as we expect this feature tohave the most pronounced impact on the band dispersion as measured with photoe-mission. The ordering vector is short compared to the size of the graphene BZ, butit is apparent that the crossings appear along the ΓK high-symmetry directions (seeFig. 4.10a). Looking at the symmetrized ARPES energy distribution curves alongthis direction (Fig. 4.10b), we observe a temperature dependent gap. The sym-metrization of the data facilitates the visualization of gap features, as it removesthe effects of the temperature dependent Fermi cut-off (see Fig. 4.13). The spec-tral intensity does not vanish completely inside the gap and no sharp quasi-particlepeaks are visible. Hence, the feature is best described as a pseudogap. The mo-90Figure 4.10: Band folding in Gd-intercalated graphene. (a) Band folding forgraphene and the ordering vector with the longest coherence length as detected byour REXS measurements. Tight-binding calculated Fermi surface (FS) for highly-doped pristine graphene in red and folded bands in dashed blue. The bands crossalong the ΓK direction as indicated by the arrows. The inset shows a close up lookof the ΓK direction. (b) ARPES measurements along the ΓK direction as a functionof temperature. The symmetrized energy dispersion curves reveal as pseudogapstable up to room temperature. (c) FS with overlaid prominent scattering vectorsas shown in Fig. 4.4d in red, green, blue, and yellow, respectively. The red arrowsconnect non-equivalent points on the FS from the M point to the ΓK direction.91Figure 4.11: Pseudogap anisotropy at room temperature. In agreement withthe prediction from the band folding, the pseudogap is mostly centered around theΓK direction. The data was taken at 300 K with measurements between the ΓMand ΓK directions (indicated by the red area in the schematic BZ in the corner).The data was then symmetrized.Figure 4.12: Pseudogap at the M point. Symmetrized ARPES energy distributioncurves (EDCs) at the M point show no pseudogap at 300 K. At lower temperatures,a pseudogap also opens up here.92mentum distribution of the gap is shown in Fig. 4.11. The largest pseudogaps arefound around the ΓK direction, as expected from the simple band folding model.This hints at a situation where the potential induced by the gadolinium orderingis not strong enough to induce measurable spectral weight on the folded bands,but nevertheless a finite coupling leads to the opening of a hybridization gap at thecrossing points with the original bands, which is then observable in photoemission.The data was taken at 300 K. Going to lower temperatures, the picture changes andwe observe a pseudogap at the M point as well (see Fig. 4.12). This is not unex-pected, as our model is based only on the scattering vector with the longest coher-ence length and hence presumably the greatest impact on the electronic dispersion.When the temperature is reduced, the coupling to other scattering vectors (compareFig. 4.4) cannot be ignored and the band folding model becomes appreciably morecomplicated with band crossings at various points of the Brillouin zone.Next, we superimpose the prominent scattering vectors found in our REXS dataon the FS of our material as determined with ARPES (Fig. 4.10c). The majority ofthe vectors (yellow, green, blue) connect equivalent points on the FS, but one vectoralong the SiC direction (red) connects the dispersion along ΓK and the M pointof two neighboring BZs. This could be connected to the suppression of spectralweight around the M point in the photoemission data. We note here that none ofthe vectors satisfies a nesting condition of the FS, a property that has been discussedextensively for various materials [280–283].Lastly, we want to look more closely at the prominent kink feature around250 meV binding energy in the ARPES measurements that connects to a “replica”band along the KMK direction (see Figs. 4.2b+c). A close up ARPES data set takenroughly along the KMK direction is shown in Fig. 4.14a. The data clearly showsthe anticrossing of two branches in the electronic dispersion. The energy distribu-tion curves (EDCs) can be fitted with two peaks to gain more insight (see Fig. 4.14).The results are summarized in Fig. 4.15. Looking at the peak positions, we canidentify the two branches of the dispersion that asymptotically approach a bind-ing energy value of about 230 meV on either side of the crossing (see Fig. 4.15a).The widths and areas of the two peaks show a similar complementary behaviourfor the upper and lower branches and indicate a shift of spectral weight from onebranch to the other across the kink as observed in the ARPES measurements (see93Figure 4.13: Symmetrization of photoemission data. An energy dispersioncurve measured with photoemission on polycrystalline gold at a temperature ofabout 10 K is shown (red circles). Photoemission only measures occupied states,hence a Fermi cut-off is visible. For the shown data the Fermi level lies at about16.85 eV kinetic energy. A Fermi function is fitted to the data around the cut-offfor illustration (blue). For symmetrization the data are first mirrored with respectto the Fermi energy (red triangles for the data and grey line for the Fermi fit) andthen the original data and the mirrored data are added (solid red diamonds for thedata and black line for the Fermi functions). The sum of the two Fermi functionsis just a flat line and polycrystalline gold is a simple metal and thus does not showany gap features at the Fermi level.Figs. 4.15b+c). Now the questions is: what is the underlying physics causing theapparent anticrossing? The two obvious candidates are coupling to a bosonic mode(e.g. a phonon mode) and the band folding discussed above.First, we look at the possibility of an interaction between a phonon mode andthe electronic dispersion of graphene. Kink features in graphene have already beendiscussed extensively in terms of electron-phonon coupling in the past and the op-tical phonon mode around 250 meV is known to couple strongly to the electronsin various graphene sample systems [62, 66, 284, 285]. Also the effects of so-94Figure 4.14: Fitting of energy distribution curves (EDCs) around ARPES kinkfeature. (a) Close up ARPES map of the kink feature along the KMK direction(compare 4.2b and 4.2c). The Fermi level is indicated by the dashed white line. (b)EDCs extracted from (a) are fitted to two peaks and a background (red lines). Therange of the fits is indicated by the length of the red lines and excludes the Fermicut-off. A clear anticrossing is observed.Figure 4.15: Fitting results of the energy distribution curves (EDCs) aroundthe ARPES kink feature. (a) Position of the two Gaussian peaks for different EDCs.The lower branch is shown in red and the upper branch in black. Both branchesasymptotically approach a binding energy value of about 230 meV (dashed blueline) for higher or lower momenta away from the kink respectively. (b) Widths ofthe two peaks around the kink feature, with the upper branch in black and the lowerbranch in red. (c) The areas of the two peaks around the kink feature show a shiftof spectral weight from one branch to the other across the anticrossing. The upperbranch is shown in black and the lower branch in red.95Figure 4.16: Simulation of coupling to a mode. (a)–(c) Simulation of the cross-ing of two modes. The linear dispersion (red) could, for example, be the electronicdispersion of graphene. The flat mode (red) could stand for an optical phononmode of graphene. (a) No coupling between the modes. (b) Coupling of 0.02 eVbetween the two modes. (c) Coupling of 0.05 eV between the two modes. (c)–(f)Same dispersions as in (a)–(c), but with a Gaussian broadening to indicate a finitelifetime of the modes and resolution effects. (g)–(i) Same dispersions and couplingconstants, but projected only on the linear dispersing (electronic) mode. This couldrepresent a simple model for a possible ARPES measurement.called “replica” bands in ARPES due to strong electron-phonon coupling have beenobserved, especially in SrTiO3 based structures [286, 287], while they were con-troversial in highly-doped graphene samples [65, 288]. To simulate the effect of acoupling between electrons and phonons, we employ a simple model with a linearlydispersing band representing the electrons and a flat band representing an opticalphonon mode (see Fig. 4.16). If we now increase the coupling between the twomodes by adjusting the off-diagonal elements in the Hamiltonian, we observe twoeffects. Firstly, with increasing coupling, an increasing gap at the point of crossing96between the two bands is observed (see Figs. 4.16a–c). Secondly, with increasingcoupling, there is an increased intermixing between the phonon and electron char-acter of the system around the crossing point. Since ARPES is only sensitive tothe electronic part of the spectrum, we can project the eigenvectors of the systemonto the electronic character only, to obtain a better representation of a possibleARPES measurement (see Figs. 4.16g–i). We see that for the right choice of cou-pling strength, we can indeed simulate the expected kink feature (Fig. 4.16h). Itshould be noted that especially for larger couplings, our simple model breaks downand more sophisticated calculations have to be conducted [289, 290]. They lead tothe appearance of higher harmonics for the phonon excitations, which have alsobeen observed experimentally [287]. Our ARPES data on Gd-intercalated grapheneshows no signs of such multi-phonon excitations.As a second possibility, we look at the band folding as an origin of the replicaband and anticrossing in the ARPES data. We start with the band structure for pris-tine graphene from a tight binding model (Fig. 4.17a). An iso-energy contour canbe selected that resembles the Fermi surface of Gd-intercalated graphene, justify-ing the choice of a band structure based on pristine graphene as a starting point(Fig. 4.17b). Now we apply the band folding based on the REXS data as alreadydescribed above (compare Fig. 4.10a), leading to a total of seven modes in the sys-tem (the original band plus six scattered bands). Due to the comparatively shortscattering vector and hence close proximity of all bands, a finite coupling leadsto a strong intermixing of the original band dispersion onto all modes. This is il-lustrated in Figs. 4.17c–i. The results of the band folding for both high-symmetrydirections KMK and KΓK are shown in Figs. 4.18a+b without coupling and inFigs. 4.18c+d with a coupling of 30 meV. The folding model can indeed produce a“replica” band around the M point as well as shift spectral weight between differentmodes as observed with ARPES (see Fig. 4.18c).Between the high-symmetry points K, M, and Γ the dispersions of all modesrun basically in parallel and are hard to distinguish for finite couplings. In anattempt to find signatures of the modes, we conduct a linewidth analysis of mo-mentum distribution curves (MDCs) along the ΓK direction (see Fig. 4.19). Fittingthe MDCs to a single Lorentzian reveals additional spectral weight on either sideof the “main” band (see Fig. 4.19a). This offset is systematic and and present along97Figure 4.17: Simulation of band folding in Gd-intercalated graphene. (a) Iso-energy contours from a tight-binding model for pristine graphene. (b) A contouris selected that most closely represents the doping level (Fermi surface) in Gd-intercalated graphene. (c)–(i) The bands are folded according to the measuredREXS scattering vector (see 4.4c and 4.10a) with a six-fold symmetry. For a fi-nite coupling (here 0.03 eV) the character of the original band gets mixed withthe folded bands. The amount of character of the graphene dispersion is shown ascolor plots for the original band (c) and the folded bands (d)–(i), indicating a strongintermixing even for moderate couplings.98Figure 4.18: Results of simulation of band folding in Gd-intercalatedgraphene. (a) Original band dispersion for pristine graphene from a tight-bindingmodel along the KMK direction (red) and folded bands (blue) without coupling.(b) Original band dispersion for pristine graphene from a tight-binding model alongthe KΓK direction (red) and folded bands (blue) without coupling. (c) Same dis-persion as in (a), but with a finite coupling of 0.03 eV and a Gaussian broadening.(d) Same dispersion as in (b), but with a finite coupling of 0.03 eV and a Gaussianbroadening.the whole analysis window (see Fig. 4.19b). Fitting the MDCs to three Lorentzianpeaks (see Fig. 4.19c) captures the additional spectral weight. Thus, this couldbe taken as further evidence for the band folding model. It should be noted thatthe folding model also leads to a splitting of the bands around the K points (seeFigs. 4.18c+d). This is not visible in the ARPES measurements, possibly due to thelarge binding energy of the Dirac point and hence energetic distance to the Fermilevel.99Figure 4.19: Analysis of Lorentzian line fits to momentum distribution curves(MDCs). (a) Exemplary MDC fitted to a constant background and a singleLorentzian (green). The fit underestimates the dispersion on either side of the peak.(b) Difference of the experimental MDCs along the ΓK direction and fits with a sin-gle Lorentzian peak. The fit systematically deviates from the experimental data. (c)When the MDCs are fitted to three instead of one Lorentzian, the additional spec-tral weight on either side of the main peak is captured, indicating the possibilityof additional bands dispersing parallel to the original one as predicted by the bandfolding simulation.In summary, we found good arguments for either possibility for the kink featureand “replica” band observed in the ARPES data. While a definite distinction cannotbe made at this point, it may be worth putting forward the idea of a feedbackbetween both mechanisms. This way, the energy scale set by the phonon mode –coupling to the electronic degrees of freedom – would make the system susceptibleand would feed back to ordering phenomena and their scattering vectors.Concluding, we have investigated a novel quantum material playground com-bining flat bands in graphene coupled to an ordered lattice of intercalated gadolin-ium atoms. We observe an interplay of quantum phases at temperatures above300 K in a purely two-dimensional and wafer-scale material. Our results stronglyhint at an intimate relationship between the observation of a pseudogap in photoe-mission and ordering phenomena detected by resonant X-ray scattering, sheddingnew light on the vastly important field of correlated electron systems outside thescope of Mott physics. We expect the material system to play an important role inthe study of highly-doped graphene, which has already been extensively discussedtheoretically in terms of spin and charge density wave orders as well as unconven-tional superconductivity in proximity to the Van Hove singularity [81, 291, 292]100Figure 4.20: Predicted phase diagram for highly-doped graphene. Predictedphase diagram of highly-doped graphene as calculated by the authors of [291].Around the Van Hove singularity, a competition between unconventional super-conductivity and a spin-density wave order is expected as function of electronicdoping.(see Fig. 4.20). Comparing the experimental findings on gadolinium intercalatedgraphene so far with the proposed phase diagram for highly-doped graphene it ap-pears that we can indeed reach the required doping levels to reach the Van Hovesingularity at the M point of the dispersion in graphene. In the phase diagram, thiswould put us in a regime where a spin density wave should be the dominating orderparameter (yellow region in Fig. 4.20). We find evidence for density wave orders inour X-ray scattering experiments, but it should be noted that the calculations weredone for pristine graphene and especially the X-ray scattering technique is mostsensitive to the gadolinium atoms. Nevertheless, combing all the experimental evi-dence it might be fruitful to think about future experiments that allow the tuning ofthe chemical potential around the Van Hove singularity to search for possible su-perconducting regions in the phase diagram. This could for example be attemptedby the deposition of additional atoms or by back gating the samples. We also wantto stress the prospect of adding magnetic interactions to the growing field of flatbands in graphene.101Chapter 5Conclusion and outlookWe have shown that monolayer graphene can provide a two-dimensional platformfor the design and investigation of novel and emergent quantum phases. The sam-ple systems are chemically simple, clean, and non-toxic. The presented examplesin this thesis display quantum phenomena on a wafer-scale platform up to roomtemperature, which can facilitate the future exploitation of the observed effects indevices and applications.In the first example, we utilize heteroepitaxial strain between graphene and thesupporting SiC substrate to induce pseudomagnetic fields. These fields are uniquefor linearly dispersing Dirac electrons and arise only for certain strain geometries.The homogeneity of the pseudomagnetic field allows us to observe the pseudo-quantum Hall effect with its Landau level quantization using angle-resolved pho-toemission spectroscopy (ARPES). Additionally, the peculiar behavior of the zerothLandau level gives us new insight into the sublattice symmetry breaking Semenoffmass term in the graphene on SiC sample system.In the second example, we show that through the intercalation of gadoliniumatoms between the SiC substrate and the monolayer graphene, we can achievethe ultra-high doping levels required to reach the Van Hove singularity at the Mpoint of the dispersion. Here a transition of the Fermi surface topology from twoelectron pockets to a single hole pocket takes place and a strong renormalizationfrom the expected band structure due to electronic correlations is observed. Thetemperature- and momentum-dependent pseudogap detected with ARPES can be102connected to ordering phenomena as determined by resonant energy-integrated X-ray scattering (REXS). The possibility of adding magnetic properties to the field offlat bands in graphene is discussed.Looking ahead to future potential experimental and theoretical research direc-tions using graphene as a versatile design platform with a range of available controlparameters, a number of possibilities appear feasible. A more tunable approach forthe creation of different strain patterns (magnitude as well as geometry), eitherthrough optimization of the growth process or top-down techniques, would enablefurther studies in the area of pseudomagnetic fields. On top of that, strain hasalso been discussed in terms of an enhancement of electron-phonon coupling andpossible resulting superconductivity [293–295]. For the gadolinium-intercalatedgraphene samples, a further study of the phononic interactions with the observedordering phenomena and a possible connection to the magnetic properties of thesample system would be beneficial. Here, theoretical efforts are hindered by thecomputational cost of the large unit cell, but resonant inelastic X-ray scattering(RIXS) experiments could help elucidate the situation. In contrast to its energy-integrated counter part REXS, RIXS also analyzes the energy of the outgoing pho-ton and – in combination with a new generation of high-resolution spectrometers– can distinguish magnetic and lattice excitations and possibly their interactions inthe sample [296–300]. It could also be feasible to directly image the spin textureof the sample using spin-polarized tips in scanning probe techniques [301–303].Going beyond the scope of this thesis, the twist angle between different gra-phene layers as a control parameter has recently received a lot of attention, follow-ing the experimental discovery of correlated electron phases and unconventionalsuperconductivity at so-called “magic” angles [74, 83, 304]. Additionally, at atwist angle of 30◦ between two graphene layers, the physics of quasicrystals with-out translational periodicity can be studied [305, 306]. Also, combining the twistangle with other control parameters appears to be a promising direction. Using hy-drostatic pressure to change the coupling between two twisted layers can tune thesuperconducting transition without the need for tedious, precise alignment of thegraphene flakes [304]. Twisting can also be combined with the strain-induced pseu-domagnetic fields discussed in this thesis. When graphene is rotated with respectto black phosphorus, the resulting Moire´ pattern leads to strain that induces pseu-103Figure 5.1: Combining twisting and strain in graphene flakes. (a) Schematicof the measurement set-up showing the STM tip. G and BP represent monolayergraphene and the multilayer black phosphorus flake respectively, SiO2 is 300 nm-thick silicon dioxide, Si is highly-doped silicon and A is the tunnelling current. Thegraphene is grounded via a Au electrode. A back-gate voltage Vg is applied throughthe doped Si electrode. (b) Sketch of graphene on BP showing the emergence ofMoire´ patterns. The rotation angle θ is defined as the angle between the BP zigzagdirection and the nearest graphene zigzag direction. (c) dI/dV spectra taken onsamples with different twist angles between graphene and black phosphorus. Aseries of Landau level peaks are visible for all samples with the first Landau levellabeled by grey circles. (d) The peak positions are plotted versus sgn(N)√|N| toshow the expected linear behaviour for Landau levels. The slope of the curves andhence the strength of the pseudomagnetic field varies with the twist angle. (e) Thestrength of the pseudomagnetic field (BS) is plotted as a function of the twist angleθ between graphene and black phosphorus, showing the tunability of the approach.The figure was adapted from [224].104domagnetic fields in graphene [224]. Depending on the twist angle, the magnitudeof the strain and hence the strength of the pseudomagnetic field can be tuned (seeFig. 5.1). Unfortunately, all of these sample systems are currently limited to smallgraphene flakes and device geometries. This makes it difficult for conventionalARPES with large spot sizes of the incident light to directly measure the electronicband structure. Fortunately, new developments in the ARPES community, basedeither on synchrotron radiation and focusing zone plates or laser-based systems inwhich optical setups can be used to reduce the spot size on the sample, can add anadditional layer of lateral control to the technique of ARPES [307–309]. These ad-vances could also benefit experiments on graphene nanoribbons. They further adddimensionality as a control knob in graphene and introduce the prospect of tailoringedge states with interesting magnetic and topological properties [67, 68, 87, 88].Finally, ultra-fast pump-probe setups in combination with ARPES experiments arebecoming more widely available. They can be used to study the dynamics of carri-ers in materials on very short time scales [94, 310, 311], generate and manipulateultra-fast currents [312, 313], induce or enhance quantum phenomena like super-conductivity [314–316], and access new physics like so-called time crystals, lim-ited strictly to non-equilibrium conditions [317]. Exciting times still lie ahead forthe platform of graphene and beyond.105Bibliography[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.Geim. The electronic properties of graphene. Reviews of Modern Physics,81(1):109–162, Jan. 2009. doi:10.1103/RevModPhys.81.109. URLhttps://link.aps.org/doi/10.1103/RevModPhys.81.109. → pages1, 13, 48, 56, 61, 77[2] A. K. Geim and K. S. Novoselov. The rise of graphene. Nature materials, 6(3):183–191, 2007. URLhttp://www.nature.com/nmat/journal/v6/n3/abs/nmat1849.html. → pages48, 56, 77[3] M. J. Allen, V. C. Tung, and R. B. Kaner. Honeycomb Carbon: A Reviewof Graphene. 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URLhttps://www.nature.com/articles/nature16522. → page 105[317] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith,G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and147C. Monroe. Observation of a discrete time crystal. Nature, 543(7644):217–220, Mar. 2017. ISSN 1476-4687. doi:10.1038/nature21413. URLhttps://www.nature.com/articles/nature21413. → page 105148Appendix APublicationsDuring the course of this PhD thesis I was involved in a number of researchprojects. Some of these projects have resulted in peer reviewed publications, whichare listed here.• Room temperature strain-induced Landau levels in graphene on a wafer-scale platform.P. Nigge, A. C. Qu, E´. Lantagne-Hurtubise, E. Ma˚rsell, S. Link, G. Tom, M.Zonno, M. Michiardi, M. Schneider, S. Zhdanovich, G. Levy, U. Starke, C.Gutie´rrez, D. Bonn, S. A. Burke, M. Franz, and A. Damascellisubmittted, arXiv:1902.00514 [cond-mat.mtrl-sci]• Correlated electron physics in gadolinium intercalated graphene.P. Nigge, A. C. Qu, S. Link, F. Boschini, J. Geurs, M. Schneider, S. Zh-danovich, G. Levy, J. Smet, R. J. Green, U. Starke, and A. Damascelliin preparation• Bandgap opening in graphene by selective symmetry breaking.A. C. Qu, P. Nigge, C. Gutie´rrez, S. Link, G. Levy, M. Michiardi, M. Schnei-der, S. Zhdanovich, U. Starke, and A. Damascelliin preparation149• Evidence for superconductivity in Li-decorated monolayer graphene.B. M. Ludbrook, G. Levy, P. Nigge, M. Zonno, M. Schneider, D. J. Dvorak,C. N. Veenstra, S. Zhdanovich, D. Wong, P. Dosanjh, C. Straßer, A. Sto¨hr,S. Forti, C. R. Ast, U. Starke, and A. DamascelliProceedings of the National Academy of Sciences. 2015 Sep 22;112(38):11795-9.• Collapse of superconductivity in cuprates via ultrafast quenching of phasecoherence.F. Boschini, E. H. da Silva Neto, E. Razzoli, M. Zonno, S. Peli, R. P. Day,M. Michiardi, M. Schneider, B. Zwartsenberg, P. Nigge, R. D. Zhong, J.Schneeloch, G. D. Gu, S. Zhdanovich, A. K. Mills, G. Levy, D. J. Jones, C.Giannetti, and A. DamascelliNature materials. 2018 May;17(5):416.• Vanishing of the pseudogap in electron-doped cuprates via quenching of thespin-correlation length.F. Boschini, M. Zonno, E. Razzoli, R. P. Day, M. Michiardi, B. Zwartsen-berg, P. Nigge, M. Schneider, E. H. da Silva Neto, A. Erb, S. Zhdanovich,A. K. Mills, G. Levy, C. Giannetti, D. J. Jones, and A. Damascellisubmitted, arXiv:1812.07583 [cond-mat.str-el]Additional publications based on work before the Phd:• Tropospheric chemistry of internally mixed sea salt and organic particles:Surprising reactivity of NaCl with weak organic acids.A. Laskin, R. C. Moffet, M. K. Gilles, J. D. Fast, R. A. Zaveri, B. Wang,P. Nigge, and J. ShutthanandanJournal of Geophysical Research: Atmospheres. 2012 Aug 16;117(D15).• An environmental sample chamber for reliable scanning transmission x-raymicroscopy measurements under water vapor.150S. T. Kelly, P. Nigge, S. Prakash, A. Laskin, B. Wang, T. Tyliszczak, S. R.Leone, and M. K. GillesReview of Scientific Instruments. 2013 Jul 30;84(7):073708.• Adsorption geometry and electronic structure of iron phthalocyanine on Agsurfaces: A LEED and photoelectron momentum mapping study.V. Feyer, M. Graus, P. Nigge, M. Wießner, R. G. Acres, C. Wiemann, C. M.Schneider, A.Scho¨ll, and F. ReinertSurface science. 2014 Mar 1;621:64-8.• The geometric and electronic structure of TCNQ and TCNQ+Mn on Ag(001) and Cu (001) surfaces.V. Feyer, M. Graus, P. Nigge, G. Zamborlini, R. G. Acres, A. Scho¨ll, F.Reinert, and C. M. SchneiderJournal of electron spectroscopy and related phenomena. 2015 Oct 1;204:125-31.• Three-dimensional tomographic imaging of molecular orbitals by photoelec-tron momentum microscopy.M. Graus, C. Metzger, M. Grimm, P. Nigge, V. Feyer, A. Scho¨ll, and F.ReinertThe European Physical Journal B. 2019 Apr 1;92(4):80.151Appendix BConference contributionsDuring the course of this PhD thesis I was fortunate to present our work during anumber of international research conferences and meetings, which are listed belowin chronological order.• Graphene and related materials: Properties and Applications (GM), May2016, invited talk (Adatom-induced superconductivity in monolayer gra-phene), Paestum, Italy• Spectroscopies in novel superconductors (SNS), June 2016, talk + poster(Adatom-induced superconductivity in monolayer graphene), Ludwigsburg,Germany• Center for quantum materials meeting (MPI-QMI-UT), December 2016, talk(Adatom-induced superconductivity in monolayer graphene), Tokyo, Japan• Canadian-American-Mexican Graduate student physics conference (CAM),August 2017, talk (Adatom-induced superconductivity in monolayer gra-phene), Washington D.C., USA• Superconductivity in atomically thin materials and heterostructures (Super-Thin), November 2017, invited talk (From evidence of superconductivity to-wards correlated electron physics in monolayer graphene), Lugano, Switzer-land152• Center for quantum materials meeting (MPI-QMI-UT), December 2017, talk(Correlated electron physics in graphene), Stuttgart, Germany• Center for quantum materials winterschool (MPI-QMI-UT), February 2018,poster (Momentum-resolved landau quantization via strain-induced pseudo-magnetic fields in graphene), Tokyo, Japan• Center for quantum materials meeting (MPI-QMI-UT), December 2018, talk(Designing quantum phases in monolayer graphene – Momentum-resolvedLandau levels in strained graphene), Tokyo, Japan• International winterschool on electronic properties of novel materials (IWE-PNM), March 2019, poster (Direct observation of momentum-resolved Lan-dau levels in strained single-layer graphene), Kirchberg, Austria• Canadian graduate quantum conference (CGQC), June 2019, poster (Mom-entum-resolved Landau quantization via strain-induced pseudomagnetic fie-lds in graphene), Sherbrooke, Canada• Properties, Fabrication and Applications of Nano-Materials and Nano-Devi-ces (Nano-M&D), June 2019, invited talk (declined), Paestum, Italy• Gordon Research Conference – New materials and structures in topologi-cal and correlated systems (GRC), June 2019, poster (Momentum-resolvedLandau quantization via strain-induced pseudomagnetic fields in graphene),Hongkong, China153

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